mathematics - sweet formula - math resources

DEPED COPY

10

Mathematics

Department of Education Republic of the Philippines

This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected].

We value your feedback and recommendations.

Learner’s ModuleUnit 2

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

mailto:[email protected]

DEPED COPY

Mathematics – Grade 10 Learner’s Module First Edition 2015

Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.

Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seeking permission to use these materials from their respective copyright owners. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them. Only institution and companies which have entered an agreement with FILCOLS and only within the agreed framework may copy this Learner’s Module. Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publisher and authors directly. Authors and publishers may email or contact FILCOLS at [email protected] or (02) 439-2204, respectively.

Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD

Printed in the Philippines by REX Book Store

Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 5th Floor Mabini Building, DepEd Complex

Meralco Avenue, Pasig City Philippines 1600

Telefax: (02) 634-1054, 634-1072 E-mail Address: [email protected]

Development Team of the Learner’s Module

Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and Rosemarievic Villena-Diaz, PhD

Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida

Editor: Maxima J. Acelajado, PhD

Reviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd, Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A. Tudlong, PhD

Illustrator: Cyrell T. Navarro

Layout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. Pagulayan

Management and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr.


mailto:[email protected]

DEPED COPY

Introduction

This material is written in support of the K to 12 Basic Education

Program to ensure attainment of standards expected of students. In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants. There are eight (8) modules in this material. Module 1 – Sequences Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills.


DEPED COPY

Unit 2

Module 3: Polynomial Functions ............................................................ 99 Lessons and Coverage ........................................................................ 100 Module Map ......................................................................................... 100 Pre-Assessment .................................................................................. 101 Learning Goals and Targets ................................................................ 105

Activity 1 .................................................................................... 106 Activity 2 .................................................................................... 107 Activity 3 .................................................................................... 108 Activity 4 .................................................................................... 108 Activity 5 .................................................................................... 110 Activity 6 .................................................................................... 111 Activity 7 .................................................................................... 112 Activity 8 .................................................................................... 115 Activity 9 .................................................................................... 115 Activity 10 .................................................................................. 118 Activity 11 .................................................................................. 119 Activity 12 .................................................................................. 121 Activity 13 .................................................................................. 122 Activity 14 .................................................................................. 123

Summary/Synthesis/Generalization ........................................................... 125 Glossary of Terms ...................................................................................... 125 References Used in this Module ................................................................. 126

Module 4: Circles .................................................................................... 127 Lessons and Coverage ........................................................................ 127 Module Map ......................................................................................... 128 Pre-Assessment .................................................................................. 129 Learning Goals and Targets ................................................................ 134

Lesson 1A: Chords, Arcs, and Central Angles .......................................... 135 Activity 1 .................................................................................... 135 Activity 2 .................................................................................... 137 Activity 3 .................................................................................... 138 Activity 4 .................................................................................... 139 Activity 5 .................................................................................... 150 Activity 6 .................................................................................... 151 Activity 7 .................................................................................... 151 Activity 8 .................................................................................... 152 Activity 9 .................................................................................... 152 Activity 10 .................................................................................. 155 Activity 11 .................................................................................. 155 Activity 12 .................................................................................. 157 Activity 13 .................................................................................. 159

Table of Contents


DEPED COPY

Summary/Synthesis/Generalization ........................................................... 160

Lesson 1B: Arcs and Inscribed Angles ....................................................... 161 Activity 1 .................................................................................... 161 Activity 2 .................................................................................... 162 Activity 3 .................................................................................... 163 Activity 4 .................................................................................... 164 Activity 5 .................................................................................... 167 Activity 6 .................................................................................... 168 Activity 7 .................................................................................... 169 Activity 8 .................................................................................... 170 Activity 9 .................................................................................... 172 Activity 10 .................................................................................. 174 Activity 11 .................................................................................. 175 Activity 12 .................................................................................. 176


Lesson 2A: Tangents and Secants of a Circle ............................................ 178 Activity 1 .................................................................................... 178 Activity 2 .................................................................................... 179 Activity 3 .................................................................................... 180 Activity 4 .................................................................................... 188 Activity 5 .................................................................................... 189 Activity 6 .................................................................................... 192 Activity 7 .................................................................................... 194 Activity 8 .................................................................................... 197


Lesson 2B: Tangent and Secant Segments ................................................. 199 Activity 1 .................................................................................... 199 Activity 2 .................................................................................... 200 Activity 3 .................................................................................... 200 Activity 4 .................................................................................... 201 Activity 5 .................................................................................... 204 Activity 6 .................................................................................... 205 Activity 7 .................................................................................... 206 Activity 8 .................................................................................... 207 Activity 9 .................................................................................... 208 Activity 10 .................................................................................. 210

Summary/Synthesis/Generalization ........................................................... 211 Glossary of Terms ....................................................................................... 212 List of Theorems and Postulates on Circles ............................................... 213 References and Website Links Used in this Module .................................. 215

Module 5: Plane Coordinate Geometry ............................................... 221 Lessons and Coverage ........................................................................ 222 Module Map ......................................................................................... 222 Pre-Assessment ................................................................................... 223 Learning Goals and Targets ................................................................. 228


DEPED COPY

Lesson 1: The Distance Formula, the Midpoint Formula,

and the Coordinate Proof .......................................................... 229 Activity 1 .................................................................................... 229 Activity 2 .................................................................................... 230 Activity 3 .................................................................................... 231 Activity 4 .................................................................................... 232 Activity 5 .................................................................................... 241 Activity 6 .................................................................................... 242 Activity 7 .................................................................................... 242 Activity 8 .................................................................................... 243 Activity 9 .................................................................................... 245 Activity 10 .................................................................................. 248 Activity 11 .................................................................................. 250


Lesson 2: The Equation of a Circle ............................................................ 252 Activity 1 .................................................................................... 252 Activity 2 .................................................................................... 253 Activity 3 .................................................................................... 254 Activity 4 .................................................................................... 263 Activity 5 .................................................................................... 265 Activity 6 .................................................................................... 265 Activity 7 .................................................................................... 266 Activity 8 .................................................................................... 267 Activity 9 .................................................................................... 267 Activity 10 .................................................................................. 269

Summary/Synthesis/Generalization ........................................................... 270 Glossary of Terms ...................................................................................... 270 References and Website Links Used in this Module ................................. 271


DEPED COPY

99

I. INTRODUCTION

You are now in Grade 10, your last year in junior high school. In this level and in the higher levels of your education, you might ask the question: What are math problems and solutions for? An incoming college student may ask, “How can designers and manufacturers make boxes having the largest volume with the least cost?” And anybody may ask: In what other fields are the mathematical concepts like functions used? How are these concepts applied?

Look at the mosaic picture below. Can you see some mathematical

representations here? Give some.

As you go through this module, you are expected to define and illustrate polynomial functions, draw the graphs of polynomial functions and solve problems involving polynomial functions. The ultimate goal of this module is for you to answer these questions: How are polynomial functions related to other fields of study? How are these used in solving real-life problems and in decision making?


DEPED COPY

100

II. LESSON AND COVERAGE

This is a one-lesson module. In this module, you will learn to:

illustrate polynomial functions graph polynomial functions solve problems involving polynomial functions

Solutions of Problems Involving Polynomial Functions

Graphs of Polynomial Functions

Illustrations of Polynomial Functions

The Polynomial Functions


DEPED COPY

101

III. PRE-ASSESSMENT

Part 1

Let us find out first what you already know related to the content of this module. Answer all items. Choose the letter that best answers each question. Please take note of the items/questions that you will not be able to answer correctly and revisit them as you go through this module for self-assessment.

1. What should n be if f(x) = xn defines a polynomial function?

A. an integer C. any number B. a nonnegative integer D. any number except 0

2. Which of the following is an example of a polynomial function?

A. 134)( 3 xx

xf C. 627)( xxxf

B. 223

232)( xxxf D. 53)( 3 xxxf

3. What is the leading coefficient of the polynomial function

42)( 3 xxxf ? A. 1 C. 3 B. 2 D. 4

4. How should the polynomial function 432)( 53 xxxxf be written in standard form?

A. 432)( 53 xxxxf C. 53 324)( xxxxf B. 35 234)( xxxxf D. 423)( 35 xxxxf

5. Which of the following could be the graph of the polynomial function

1234 23 xxxy ?

A. B. C. D. x x

x

y y y y


DEPED COPY

102

6. From the choices, which polynomial function in factored form represents the given graph?

7. If you will draw the graph of 2)2( xxy , how will you sketch it with

respect to the x-axis?

A. Sketch it crossing both (-2,0) and (0,0). B. Sketch it crossing (-2,0) and tangent at (0,0). C. Sketch it tangent at (-2,0) and crossing (0,0). D. Sketch it tangent at both (-2,0) and (0,0).

8. What are the end behaviors of the graph of 432)( 53 xxxxf ?

A. rises to the left and falls to the right B. falls to the left and rises to the right C. rises to both directions D. falls to both directions

9. You are asked to illustrate the sketch of 43)( 53 xxxf using its

properties. Which will be your sketch?

A. B. C. D.

A. )1)(1)(2( xxxy

B. )2)(1)(1( xxxy

C. )1)(1)(2( xxxxy

D. )2)(1)(1( xxxxy

y

x

y

x

y

x

y

x


DEPED COPY

103

10. Your classmate Linus encounters difficulties in showing a sketch of the graph of 3 22 3 4y x x x 6. You know that the quickest technique is the Leading Coefficient Test. You want to help Linus in his problem. What hint/clue should you give?

A. The graph falls to the left and rises to the right. B. The graph rises to both left and right. C. The graph rises to the left and falls to the right. D. The graph falls to both left and right.

11. If you will be asked to choose from -2, 2, 3, and 4, what values for a

and n will you consider so that y = axn could define the graph below?

12. A car manufacturer determines that its profit, P, in thousands of pesos, can be modeled by the function P(x) = 0.00125x4 + x – 3, where x represents the number of cars sold. What is the profit when x = 300?

A. Php 101.25 C. Php 3,000,000.00 B. Php 1,039,500.00 D. Php 10,125,297.00

13. A demographer predicts that the population, P, of a town t years from

now can be modeled by the function P(t) = 6t4 – 5t3 + 200t + 12 000. What will the population of the town be two (2) years from now?

A. 12 456 C. 1 245 600 B. 124 560 D. 12 456 000

14. Consider this Revenue-Advertising Expense situation:

The total revenue R (in millions of pesos) for a company is related to its advertising expense by the function

3 21R x 600x , 0 x 400100 000

where x is the amount spent on advertising (in ten thousands of pesos).

A. a = 2 , n = 3

B. a = 3 , n = 2

C. a = - 2 , n = 4

D. a = - 2 , n = 3


DEPED COPY

104

Currently, the company spends Php 2,000,000.00 for advertisement. If you are the company manager, what best decision can you make with this business circumstance based on the given function with its restricted domain?

A. I will increase my advertising expenses to Php 2,500,000.00 because this will give a higher revenue than what the company currently earns.

B. I will decrease my advertising expenses to Php 1,500,000.00 because this will give a higher revenue than what the company currently earns.

C. I will decrease my advertising expenses to Php 1,500,000.00 because lower cost means higher revenue.

D. It does not matter how much I spend for advertisement, my revenue will stay the same.

Part 2 Read and analyze the situation below. Then, answer the question and perform the tasks that follow.

Karl Benedic, the president of Mathematics Club, proposed a project: to put up a rectangular Math Garden whose lot perimeter is 36 meters. He was soliciting suggestions from the members for feasible dimensions of the lot.

Suppose you are a member of the club, what will you suggest to Karl Benedic if you want a maximum lot area? You must convince him through a mathematical solution. Consider the following guidelines:

1. Make an illustration of the lot with the needed labels. 2. Solve the problem. Hint: Consider the formulas P = 2l + 2w for

perimeter and A = lw for the area of the rectangle. Use the formula for P and the given information in the problem to express A in terms of either l or w.

3. Make a second illustration that satisfies the findings in the solution made in number 2.

4. Submit your solution on a sheet of paper with recommendations.


DEPED COPY

105

Rubric for Rating the Output:

Score Descriptors

4 The problem is correctly modeled with a quadratic function, appropriate mathematical concepts are fully used in the solution, and the correct final answer is obtained.

3 The problem is correctly modeled with a quadratic function, appropriate mathematical concepts are partially used in the solution, and the correct final answer is obtained.

2 The problem is not properly modeled with a quadratic function, other alternative mathematical concepts are used in the solution, and the correct final answer is obtained.

1 The problem is not totally modeled with a quadratic function, a solution is presented but has incorrect final answer.

The additional two (2) points will be determined from the illustrations

made. One (1) point for each if properly drawn with necessary labels.

IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate

understanding of key concepts on polynomial functions. Furthermore, you should be able to conduct a mathematical investigation involving polynomial functions.


DEPED COPY

106

Start this module by recalling your knowledge on the concept of

polynomial expressions. This knowledge will help you understand the formal definition of a polynomial function.

Determine whether each of the following is a polynomial expression or not. Give your reasons.

1. 14x 6.

2. 35 4 2x x x 7. 3 23 3 9 2x x x 3. 2014x 8. 3 2 1x x

4. 3 14 43 7x x 9. 100 1004 4x x

5. 3 4 51 2 32 3 4x x x

10. 1 – 16x2

Did you answer each item correctly? Do you remember when an expression is a polynomial? We defined a related concept below.

A polynomial function is a function of the form

1 2

1 2 1 0...n n nn n nP x a x a x a x a x a , ,0na

where n is a nonnegative integer, naaa ...,,, 10 are real numbers called coefficients, n

nxa is the leading term, na is the leading coefficient, and 0a is the constant term.

The terms of a polynomial may be written in any order. However, if they are written in decreasing powers of x, we say the polynomial function is in standard form.

Other than P(x), a polynomial function may also be denoted by f(x). Sometimes, a polynomial function is represented by a set P of ordered pairs (x,y). Thus, a polynomial function can be written in different ways, like the following.

Activity 1:


DEPED COPY

107

f ( x )

1 2

1 2 1 0...n n nn n na x a x a x a x a

or 1 2

1 2 1 0...n n nn n ny a x a x a x a x a

Polynomials may also be written in factored form and as a product of

irreducible factors, that is, a factor that can no longer be factored using coefficients that are real numbers. Here are some examples.

a. y = x4 + 2x3 – x2 + 14x – 56 in factored form is y = (x2 + 7)(x – 2)(x + 4) b. y = x4 + 2x3 – 13x2 – 10x in factored form is y = x(x – 5)(x + 1)(x + 2) c. y = 6x3 + 45x2 + 66x – 45 in factored form is y = 3(2x – 1)(x + 3)(x + 5) d. f(x) = x3 + x2 + 18 in factored form is f(x) = (x2 – 2x + 6)(x + 3) e. f(x) = 2x3 + 5x2 + 7x – 5 in factored form is f(x) = (x2 + 3x + 5)(2x – 1)

Consider the given polynomial functions and fill in the table below.

Polynomial Function Polynomial Function in

Standard Form Degree Leading

Coefficient Constant

Term

1. f ( x ) = 2 – 11x + 2x2

2. f ( x ) 32 5 153 3x x

3. y = x (x2 – 5)

4. 3 3y x x x

5. 2)1)(1)(4( xxxy

After doing this activity, it is expected that the definition of a polynomial

function and the concepts associated with it become clear to you. Do the next activity so that your skills will be honed as you give more examples of polynomial functions.

Activity 2:


DEPED COPY

108

Use all the numbers in the box once as coefficients or exponents to form as many polynomial functions of x as you can. Write your polynomial functions in standard form.

1 –2 47

2 61

3

How many polynomial functions were you able to give? Classify each

according to its degree. Also, identify the leading coefficient and the constant term.

In this section, you need to revisit the lessons and your knowledge on evaluating polynomials, factoring polynomials, solving polynomial equations, and graphing by point-plotting. Your knowledge of these topics will help you sketch the graph of polynomial functions manually. You may also use graphing utilities/tools in order to have a clearer view and a more convenient way of describing the features of the graph. Also, you will focus on polynomial functions of degree 3 and higher, since graphing linear and quadratic functions were already taught in previous grade levels. Learning to graph polynomial functions requires your appreciation of its behavior and other properties.

Factor each polynomial completely using any method. Enjoy working with your seatmate using the Think-Pair-Share strategy.

1. (x – 1) (x2 – 5x + 6) 2. (x2 + x – 6) (x2 – 6x + 9) 3. (2x2 – 5x + 3) (x – 3) 4. x3 + 3x2 – 4x – 12 5. 2x4 + 7x3 – 4x2 – 27x – 18

Did you get the answers correctly? What method(s) did you use? Now, do the same with polynomial functions. Write each of the following polynomial functions in factored form:

Activity 4:

Activity 3:


DEPED COPY

109

6. xxxy 1223 7. 164 xy 8. 68482 234 xxxxy 9. xxxy 910 35

10. 1827472 234 xxxxy

The preceding task is very important for you since it has something to do with the x-intercepts of a graph. These are the x-values when y = 0, thus, the point(s) where the graph intersects the x-axis can be determined.

To recall the relationship between factors and x-intercepts, consider

these examples:

a. Find the intercepts of 64 23 xxxy . Solution: To find the x-intercept/s, set y = 0. Use the factored form. That is, y = x3 – 4x2 + x + 6

y = (x + 1)(x – 2)(x – 3) Factor completely. 0 = (x + 1)(x – 2)(x – 3) Equate y to 0.

x + 1 = 0

x = –1 or x – 2 = 0

x = 2 or x – 3 = 0

x = 3

The x-intercepts are –1, 2, and 3. This means the graph will pass through (-1, 0), (2, 0), and (3, 0).

Finding the y-intercept is more straightforward. Simply set x = 0

in the given polynomial. That is,

y = x3 – 4x2 + x + 6 y = 03 – 4(0)2 + 0 + 6 y = 6

The y-intercept is 6. This means the graph will also pass through (0,6).

Equate each factor to 0 to determine x.


DEPED COPY

110

b. Find the intercepts of xxxxy 66 234

Solution: For the x-intercept(s), find x when y = 0. Use the factored form. That is, y = x4 + 6x3 – x2 – 6x

y = x(x + 6)(x + 1)(x – 1) 0 = x(x + 6)(x + 1)(x – 1)

x = 0

or x + 6 = 0 x = –6

or

x + 1 = 0 x = –1

or

x – 1 = 0 x = 1

The x-intercepts are -6, -1, 0, and 1. This means the graph will pass through (-6,0), (-1,0), (0,0), and (1,0).

Again, finding the y-intercept simply requires us to set x = 0 in the given polynomial. That is,

4 3 26 6y x x x x y (04) 6(03) (02) 6(0) 0y

The y-intercept is 0. This means the graph will pass also through (0,0).

You have been provided illustrative examples of solving for the x- and y- intercepts, an important step in graphing a polynomial function. Remember, these intercepts are used to determine the points where the graph intersects or touches the x-axis and the y-axis. But these points are not sufficient to draw the graph of polynomial functions. Enjoy as you learn by performing the next activities.

Determine the intercepts of the graphs of the following polynomial functions:

1. y = x3 + x2 – 12x 2. y = (x – 2)(x – 1)(x + 3) 3. y = 2x4 + 8x3 + 4x2 – 8x – 6 4. y = –x4 + 16 5. y = x5 + 10x3 – 9x

You have learned how to find the intercepts of a polynomial function. You will discover more properties as you go through the next activities.

Activity 5:

Equate each factor to 0 to determine x.

Factor completely. Equate y to 0.


DEPED COPY

111

Work with your friends. Determine the x-intercept/s and the y-intercept of each given polynomial function. To obtain other points on the graph, find the value of y that corresponds to each value of x in the table.

1. y = (x + 4)(x + 2)(x – 1)(x – 3) x-intercepts: __ __ __ __ y-intercept: __

x -5 -3 0 2 4 y

List all your answers above as ordered pairs.

2. y = –(x + 5)(2x + 3)(x – 2)(x – 4) x-intercepts: __ __ __ __ y-intercept: __

x -6 -4 -0.5 3 5 y

List all your answers above as ordered pairs. y = –x(x + 6)(3x – 4) x-intercepts: __ __ __

y-intercept: __

x -7 -3 1 2 y

List all your answers above as ordered pairs.

3. y = x2(x + 3)(x + 1)(x – 1)(x – 3) x-intercepts: __ __ __ __ __ y-intercept: __

x -4 -2 -0.5 0.5 2 4 y

List all your answers above as ordered pairs In this activity, you evaluated a function at given values of x. Notice that

some of the given x-values are less than the least x-intercept, some are between two x-intercepts, and some are greater than the greatest x intercept. For example, in number 1, the x-intercepts are -4, -2, 1, and 3. The value -5 is used as x-value less than -4; -3, 0, and 2 are between two x-intercepts; and 4 is used as x-value greater than 3. Why do you think we should consider them?

Activity 6:


DEPED COPY

112

In the next activity, you will describe the behavior of the graph of a polynomial function relative to the x-axis.

Given the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3), complete the table below. Answer the questions that follow.

Value of x

Value of y

Relation of y value to 0: y > 0, y = 0, or y < 0?

Location of the point (x, y): above the x-axis, on the

x-axis, or below the x-axis? -5 144 0y above the x-axis -4 -3 -2 0 y = 0 on the x - axis 0 1 2 3 4

Questions:

1. At what point(s) does the graph pass through the x-axis? 2. If 4x , what can you say about the graph? 3. If 24 x , what can you say about the graph? 4. If 12 x , what can you say about the graph? 5. If 31 x , what can you say about the graph? 6. If 3x , what can you say about the graph?

Now, this table may be transformed into a simpler one that will instantly help you in locating the curve. We call this the table of signs.

The roots of the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3) are x = –4, –2, 1, and 3. These are the only values of x where the graph will cross the x-axis. These roots partition the number line into intervals. Test values are then chosen from within each interval.

Activity 7:


DEPED COPY

113

The table of signs and the rough sketch of the graph of this function can now be constructed, as shown below. The Table of Signs

Intervals

4x 4 2x 2 1x 1 3x 3x Test value -5 -3 0 2 4

4x – + + + +

2x – – + + +

1x – – – + + 3x – – – – +

)3)(1)(2)(4( xxxxy + – + – + Position of the curve relative to the x-axis above below above below above

The Graph of )3)(1)(2)(4( xxxxy

We can now use the information from the table of signs to construct a possible graph of the function. At this level, though, we cannot determine the turning points of the graph, we can only be certain that the graph is correct with respect to intervals where the graph is above, below, or on the x-axis.

The arrow heads at both ends of the graph signify that the graph indefinitely goes upward.


DEPED COPY

114

Here is another example: Sketch the graph of )43)(2()( xxxxf

Roots of f(x): -2, 0, 34

Table of Signs:

Intervals

2x 2 0x 40

3x

43

x

Test value -3 -1 1 2

–x + + – –

x + 2 – + + +

3x – 4 – – – +

f(x) = –x(x + 2)(3x – 4) + – + – Position of the curve relative to the x-axis above below above below

Graph:

In this activity, you learned how to sketch the graph of polynomial

functions using the intercepts, some points, and the position of the curves determined from the table of signs. The procedures described are applicable when the polynomial function is in factored form. Otherwise, you need to express first a polynomial in factored form. Try this in the next activity.


DEPED COPY

115

For each of the following functions, give

(a) the x-intercept(s) (b) the intervals obtained when the x-intercepts

are used to partition the number line (c) the table of signs (d) a sketch of the graph

1. y = (2x + 3)(x – 1)(x – 4) 2. y = –x3 + 2x2 + 11x – 2 3. y = x4 – 26x2 + 25 4. y = –x4 – 5x3 + 3x2 + 13x – 10 5. y = x2(x + 3)(x + 1)4(x – 1)3

Post your answer/output for a walk-through. For each of these

polynomial functions, answer the following:

a. What happens to the graph as x decreases without bound? b. For which interval(s) is the graph (i) above and (ii) below the

x-axis? c. What happens to the graph as x increases without bound? d. What is the leading term of the polynomial function? e. What are the leading coefficient and the degree of the function?

Now, the big question for you is: Do the leading coefficient and degree

affect the behavior of its graph? You will answer this after an investigation in the next activity.

After sketching manually the graphs of the five functions given in Activity 8, you will now be shown polynomial functions and their corresponding graphs. Study each figure and answer the questions that follow. Summarize your answers using a table similar to the one provided.

Activity 9:

Activity 8:


DEPED COPY

116

Case 1 The graph on the right is defined by y = 2x3 – 7x2 – 7x + 12 or, in factored form, y = (2x + 3) (x – 1) (x – 4). Questions:

a. Is the leading coefficient a positive or a negative number?

b. Is the polynomial of even degree or odd degree?

c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right?

Case 2 The graph on the right is defined by

473 2345 xxxxy or, in factored form,

22 )2)(1()1( xxxy . Questions:





xxxy 67 24 or, in factored form, )2)(1)(3( xxxxy .

Questions:




x

y

x

y

x

y


DEPED COPY

117


2414132 234 xxxxy or, in factored form,

)4)(2)(1)(3( xxxxy . Questions:




Now, complete this table. In the last column, draw a possible graph for

the function, showing how the function behaves. (You do not need to place your graph on the xy-plane). The first one is done for you.

Sample Polynomial Function

Leading Coefficient:

0n or 0n

Degree: Even or

Odd

Behavior of the Graph:

Rising or Falling Possible Sketch Left-

hand Right-hand

1. 3 22 7 7 12y x x x 0n odd falling rising

2. 5 4 3 23 7 4y x x x x

3. 4 27 6y x x x

4. 4 3 22 13 14 24y x x x x

Summarize your findings from the four cases above. What do you observe if:

1. the degree of the polynomial is odd and the leading coefficient is

positive? 2. the degree of the polynomial is odd and the leading coefficient is

negative? 3. the degree of the polynomial is even and the leading coefficient is

positive? 4. the degree of the polynomial is even and the leading coefficient is

negative?

x

y


DEPED COPY

118

Congratulations! You have now illustrated The Leading Coefficient Test. You should have realized that this test can help you determine the end behaviors of the graph of a polynomial function as x increases or decreases without bound.

Recall that you have already learned two properties of the graph of polynomial functions; namely, the intercepts which can be obtained from the Rational Root Theorem, and the end behaviors which can be identified using the Leading Coefficient Test. Another helpful strategy is to determine whether the graph crosses or is tangent to the x-axis at each x-intercept. This strategy involves the concept of multiplicity of a zero of a polynomial function. Multiplicity tells how many times a particular number is a zero or root for the given polynomial.

The next activity will help you understand the relationship between multiplicity of a root and whether a graph crosses or is tangent to the x-axis.

Given the function )2()1()1()2( 432 xxxxy and its graph, complete the table below, then answer the questions that follow.

Root or Zero Multiplicity Characteristic of Multiplicity: Odd or Even

Behavior of Graph Relative to x-axis at this

Root: Crosses or Is Tangent to

-2 -1 1 2

Activity 10:

x

y


DEPED COPY

119

Questions:

a. What do you notice about the graph when it passes through a root of even multiplicity?

b. What do you notice about the graph when it passes through a root of odd multiplicity? This activity extends what you learned when using a table of signs to

graph a polynomial function. When the graph crosses the x-axis, it means the graph changes from positive to negative or vice versa. But if the graph is tangent to the x-axis, it means that the graph is either positive on both sides of the root, or negative on both sides of the root.

In the next activity, you will consider the number of turning points of the graph of a polynomial function. The turning points of a graph occur when the function changes from decreasing to increasing or from increasing to decreasing values.

Complete the table below. Then answer the questions that follow.

Polynomial Function Sketch Degree Number of

Turning Points

1. 4xy

2. 152 24 xxy

3. 5xy

Activity 11:

x

y

x

y x

y


DEPED COPY

120

Polynomial Function Sketch Degree Number of

Turning Points

4. 1235 xxxy

5. xxxy 45 35

Questions:

a. What do you notice about the number of turning points of the quartic functions (numbers 1 and 2)? How about of quintic functions (numbers 3 to 5)?

b. From the given examples, do you think it is possible for the degree of a function to be less than the number of turning points?

c. State the relation of the number of turning points of a function with its degree n. In this section, you have encountered important concepts that can help

you graph polynomial functions. Notice that the graph of a polynomial function is continuous, smooth, and has rounded turns. Further, the number of turning points in the graph of a polynomial is strictly less than the degree of the polynomial.

Use what you have learned as you perform the activities in the

succeeding sections.

y

x

x

y


DEPED COPY

121

The goal of this section is to help you think critically and creatively

as you apply the techniques in graphing polynomial functions. Also, this section aims to provide opportunities to solve real-life problems involving polynomial functions.

For each given polynomial function, describe or determine the following, then sketch the graph. You may need a calculator in some computations.

a. leading term b. end behaviors c. x-intercepts points on the x-axis d. multiplicity of roots e. y-intercept point on the y-axis

f. number of turning points g. sketch

1. )52()1)(3( 2 xxxy 2. 322 )2()1)(5( xxxy 3. 422 23 xxxy 4. )32)(7( 22 xxxy 5. 2861832 234 xxxxy

In this activity, you were given the opportunity to sketch the graph of

polynomial functions. Were you able to apply all the necessary concepts and properties in graphing each function? The next activity will let you see the connections of these mathematics concepts to real life.

Activity 12:


DEPED COPY

122

Work in groups. Apply the concepts of polynomial functions to answer the questions in each problem. Use a calculator when needed. 1. Look at the pictures below. What do these tell us? Filipinos need to take

the problem of deforestation seriously.

The table below shows the forest cover of the Philippines in relation to its total land area of approximately 30 million hectares.

Year 1900 1920 1960 1970 1987 1998

Forest Cover (%) 70 60 40 34 23.7 22.2

Source: Environmental Science for Social Change, Decline of the Philippine Forest A cubic polynomial that best models the data is given by

3 226 3500 391 300 69 717 000; 0 981 000 000

x x xy x

where y is the percent forest cover x years from 1900.

10 20 30 40 50 60 70 80 90 x-10

1020304050607080y

O

Activity 13:


DEPED COPY

123

Questions/Tasks: a. Using the graph, what is the approximate forest cover during the year

1940? b. Compare the forest cover in 1987 (as given in the table) to the forest

cover given by the polynomial function. Why are these values not exactly the same?

c. Do you think you can use the polynomial to predict the forest cover in the year 2100? Why or why not?

2. The members of a group of packaging designers of a gift shop are looking

for a precise procedure to make an open rectangular box with a volume of 560 cubic inches from a 24-inch by 18-inch rectangular piece of material. The main problem is how to identify the side of identical squares to be cut from the four corners of the rectangular sheet so that such box can be made. Question/Task:

Suppose you are chosen as the leader and you are tasked to lead in solving the problem. What will you do to meet the specifications needed for the box? Show a mathematical solution.

Were you surprised that polynomial functions have real and practical

uses? What do you need to solve these kinds of problems? Enjoy learning as you proceed to the next section.

The goal of this section is to check if you can apply polynomial

functions to real-life problems and produce a concrete object that satisfies the conditions given in the problem.

Read the problem carefully and answer the questions that follow.

You are designing candle-making kits. Each kit contains 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle with a square base. You want the height of the candle to be 2 inches less than the edge of the base.

Activity 14:


DEPED COPY

124

Questions/Tasks:

1. What should the dimensions of your candle mold be? Show a mathematical procedure in determining the dimensions.

2. Use a sheet of cardboard as sample material in preparing a candle mold with such dimensions. The bottom of the mold should be closed. The height of one face of the pyramid should be indicated.

3. Write your solution in one of the faces of your output (mold). Rubric for the Mathematical Solution

Point Descriptor

4 The problem is correctly modeled with a polynomial function, appropriate mathematical concepts are used in the solution, and the correct final answer is obtained.

3 The problem is correctly modeled with a polynomial function, appropriate mathematical concepts are partially used in the solution, and the correct final answer is obtained.

2 The problem is not properly modeled with a polynomial function, other alternative mathematical concepts are used in the solution, and the correct final answer is obtained.

1 The problem is not properly modeled with a polynomial function, a solution is presented but the final answer is incorrect.

Criteria for Rating the Output:

The mold has the needed dimensions and parts. The mold is properly labeled with the required length of parts. The mold is durable. The mold is neat and presentable.

Point/s to Be Given: 4 points if all items in the criteria are evident 3 points if any three of the items are evident 2 points if any two of the items are evident 1 point if any of the items is evident


DEPED COPY

125

SUMMARY/SYNTHESIS/GENERALIZATION

This lesson was about polynomial functions. You learned how to:

illustrate and describe polynomial functions; show the graph of polynomial functions using the following properties:

- the intercepts (x-intercept and y-intercept); - the behavior of the graph using the Leading Coefficient Test, table

of signs, turning points, and multiplicity of zeros; and solve real-life problems that can be modeled with polynomial functions.

GLOSSARY OF TERMS Constant Function - a polynomial function whose degree is 0 Evaluating a Polynomial - a process of finding the value of the polynomial at a given value in its domain Intercepts of a Graph - points on the graph that have zero as either the x-coordinate or the y-coordinate Irreducible Factor - a factor that can no longer be factored using coefficients that are real numbers Leading Coefficient Test - a test that uses the leading term of the polynomial function to determine the right-hand and the left-hand behaviors of the graph Linear Function - a polynomial function whose degree is 1 Multiplicity of a Root - tells how many times a particular number is a root for the given polynomial Nonnegative Integer - zero or any positive integer Polynomial Function - a function denoted by

012

21

1 ...)( axaxaxaxaxP nn

nn

nn

, where n is a nonnegative integer, naaa ...,,, 10 are real numbers called coefficients but ,0na n

nxa is the leading term, na is the leading coefficient, and 0a is the constant term Polynomial in Standard Form - any polynomial whose terms are arranged in decreasing powers of x Quadratic Function – a polynomial function whose degree is 2


DEPED COPY

126

Quartic Function – a polynomial function whose degree is 4 Quintic Function – a polynomial function whose degree is 5 Turning Point – a point where the function changes from decreasing to increasing or from increasing to decreasing values

REFERENCES USED IN THIS MODULE: Alferez, M.S., Duro, MC.A. & Tupaz, KK.L. (2008). MSA Advanced Algebra.

Quezon City, Philippines: MSA Publishing House. Berry, J., Graham, T., Sharp, J. & Berry, E. (2003). Schaum’s A-Z

Mathematics. London, United Kingdom: Hodder & Stoughton Educational.

Cabral, E. A., De Lara-Tuprio, E.P., De Las Penas, ML. N., Francisco, F. F.,

Garces, IJ. L., Marcelo, R.M. & Sarmiento, J. F. (2010). Precalculus. Quezon City, Philippines: Ateneo de Manila University Press.

Jose-Dilao, S., Orines, F. B. & Bernabe, J.G. (2003). Advanced Algebra,

Trigonometry and Statistics. Quezon City, Philippines: JTW Corporation.

Lamayo, F. C. & Deauna, M. C. (1990). Fourth Year Integrated Mathematics.

Quezon City, Philippines: Phoenix Publishing House, Inc. Larson, R. & Hostetler, R. P. (2012). Algebra and Trigonometry. Pasig City,

Philippines: Cengage Learning Asia Pte. Ltd. Marasigan, J. A., Coronel, A.C. & Coronel, I.C. (2004). Advanced Algebra

with Trigonometry and Statistics. Makati City, Philippines: The Bookmark, Inc.

Quimpo, N. F. (2005). A Course in Freshman Algebra. Quezon City,

Philippines. Uy, F. B. & Ocampo, J.L. (2000). Board Primer in Mathematics. Mandaluyong

City, Philippines: Capitol Publishing House. Villaluna, T. T. & Van Zandt, GE. L. (2009). Hands-on, Minds-on Activities in

Mathematics IV. Quezon City, Philippines: St. Jude Thaddeus Publications.


DEPED COPY

127

I. INTRODUCTION

Have you imagined yourself pushing a cart or riding in a bus having wheels that are not round? Do you think you can move heavy objects from one place to another easily or travel distant places as fast as you can? What difficulty do you think would you experience without circles? Have you ever thought of the importance of circles in the field of transportation, industries, sports, navigation, carpentry, and in your daily life?

Find out the answers to these questions and determine the vast applications of circles through this module.

II. LESSONS AND COVERAGE:

In this module, you will examine the above questions when you take the following lessons:

Lesson 1A – Chords, Arcs, and Central Angles Lesson 1B – Arcs and Inscribed Angles Lesson 2A – Tangents and Secants of a Circle Lesson 2B – Tangent and Secant Segments


DEPED COPY

128

In these lessons, you will learn to: Lesson 1A Lesson 1B

derive inductively the relations among chords, arcs, central angles, and inscribed angles;

illustrate segments and sectors of circles; prove theorems related to chords, arcs, central angles,

and inscribed angles; and solve problems involving chords, arcs, central angles,

and inscribed angles of circles. Lesson 2A Lesson 2B

illustrate tangents and secants of circles; prove theorems on tangents and secants; and solve problems involving tangents and secants of

circles.

Here is a simple map of the lessons that will be covered in this module:

Circles

Applications of Circles

Relationships among Chords, Arcs, Central Angles, and Inscribed

Angles

Tangents and Secants of Circles

Chords, Arcs, and Central

Angles

Arcs and Inscribed Angles

Tangents and Secants

Tangent and Secant

Segments


DEPED COPY

129

III. PRE-ASSESSMENT

Part I

Find out how much you already know about the topics in this module. Choose the letter that you think best answers each of the following questions. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module.

1. What is an angle whose vertex is on a circle and whose sides contain

chords of the circle? A. central angle C. circumscribed angle B. inscribed angle D. intercepted angle

2. An arc of a circle measures 30°. If the radius of the circle is 5 cm, what

is the length of the arc? A. 2.62 cm B. 2.3 cm C. 1.86 cm D. 1.5 cm

3. Using the figure below, which of the following is an external secant

segment of M?

A. CO C. NO

B. TI D. NI

4. The opposite angles of a quadrilateral inscribed in a circle are _____. A. right C. complementary B. obtuse D. supplementary

5. In S at the right, what is VSIm if mVI = 140?

A. 35 C. 140 B. 75 D. 230

6. What is the sum of the measures of the central angles of a circle with no common interior points? A. 120 B. 240 C. 360 D. 480

I N

T E

C

O

M


DEPED COPY

130

P

X

Y

N M

7. Catherine designed a pendant. It is a regular hexagon set in a circle. Suppose the opposite vertices are connected by line segments and meet at the center of the circle. What is the measure of each angle formed at the center?

A. 5.22 B. 45 C. 60 D. 72

8. If an inscribed angle of a circle intercepts a semicircle, then the angle

is _________. A. acute B. right C. obtuse D. straight

9. At a given point on the circle, how many line/s can be drawn that is

tangent to the circle? A. one B. two C. three D. four

10. What is the length of ZK in the figure on the right? A. 2.86 units C. 8 units B. 6 units D. 8.75 units

11. In the figure on the right, mXY = 150 and mMN = 30. What is XPYm ? A. 60 B. 90 C. 120 D. 180

12. The top view of a circular table shown on the right has a radius of 120 cm. Find the area of the smaller segment of the table (shaded

region) determined by a 60 arc.

A. 2400 3600 3 cm2

B. 3600 3 cm2

C. 2400 cm2

D. 14 400 3600 3 cm2

60°

120 cm


DEPED COPY

131

13. In O given below, what is PR if NO = 15 units and ES = 6 units? A. 28 units B. 24 units C. 12 units D. 9 units

14. A dart board has a diameter of 40 cm and is divided into 20 congruent

sectors. What is the area of one of the sectors?

A. 20 cm2 C. 80 cm2

B. 40 cm2 D. 800 cm2 15. Mr. Soriano wanted to plant three different colors of roses on the outer

rim of a circular garden. He stretched two strings from a point external to the circle to see how the circular rim can be divided into three portions as shown in the figure below.

What is the measure of minor arc AB? A. 64° B. 104° C. 168° D. 192°

16. In the figure below, SY and EY are secants. If SY = 15 cm,

TY = 6 cm, and LY = 8 cm. What is the length of EY ? A. 20 cm B. 12 cm C. 11.25 cm D. 6.75 cm

P

E R

O

S

N

S

Y T

L E

A

B M

C

20°

192°


DEPED COPY

132

17. In C below, mAB = 60 and its radius is 6 cm. What is the area of the shaded region in terms of pi ( )?

A. 6 cm 2 C. 10 cm 2

B. 8 cm 2 D. 12 cm 2

18. In the circle below, what is the measure of SAY if DSY is a

semicircle and ?70SADm

A. 20 C. 110 B. 70 D. 150

19. Quadrilateral SMIL is inscribed in E. If 78m SMI and

m 95MSL , find MILm .

A. 78 C. 95 B. 85 D. 102

20. In M on the right, what is BROm if 60BMOm ?

A. 120 C. 30 B. 60 D. 15

A D

S

Y

C B

A

6 cm

60°

M

I

L

78°

S 95°

E

B

M

O

R


DEPED COPY

133

Part II

Solve each of the following problems. Show your complete solutions.

2. A bicycle chain fits tightly around two gears. What is the distance between the

centers of the gears if the radii of the bigger and smaller gears are 9.3 inches and 2.4 inches, respectively, and the portion of the chain tangent to the two gears is 26.5 inches long?

Rubric for Problem Solving

Score Descriptors

4 Used an appropriate strategy to come up with the correct solution and arrived at a correct answer.

3 Used an appropriate strategy to come up with a solution, but a part of the solution led to an incorrect answer.

2 Used an appropriate strategy but came up with an entirely wrong solution that led to an incorrect answer.

1 Attempted to solve the problem but used an inappropriate strategy that led to a wrong solution.

Part III

Read and understand the situation below, then answer the questions and perform what is required. The committee in-charge of the Search for the Cleanest and Greenest School informed your principal that your school has been selected as a regional finalist. Being a regional finalist, your principal would like to make your school more beautiful and clean by making more gardens of different shapes. He decided that every year level will be assigned to prepare a garden of particular shape.

In your grade level, he said that you will be preparing circular, semicircular, or arch-shaped gardens in front of your building. He further encouraged your grade level to add garden accessories to make the gardens more presentable and amusing.

1. Mr. Javier designed an arch made of bent iron for the top of a school’s main entrance. The 12 segments between the two concentric semicircles are each 0.8 meter long. Suppose the diameter of the inner semicircle is 4 meters. What is the total length of the bent iron used to make this arch?

4 m 0.8 m


DEPED COPY

134

1. How will you prepare the design of the gardens? 2. What garden accessories will you use? 3. Make the designs of the gardens which will be placed in front of your

grade level building. Use the different shapes that were required by your principal.

4. Illustrate every part or portion of the garden including their measurements. 5. Using the designs of the gardens made, determine all the concepts or

principles related to circles. 6. Formulate problems involving these mathematics concepts or principles,

then solve.

Rubric for Design

Score Descriptors 4 The design is accurately made, presentable, and appropriate. 3 The design is accurately made and appropriate but not presentable. 2 The design is not accurately made but appropriate. 1 The design is made but not accurate and appropriate.

Rubric on Problems Formulated and Solved

Score Descriptors

6

Poses a more complex problem with two or more correct possible solutions, communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate.

5 Poses a more complex problem and finishes all significant parts of the solution, communicates ideas unmistakably, and shows in-depth comprehension of the pertinent concepts and/or processes.

4 Poses a complex problem and finishes all significant parts of the solution, communicates ideas unmistakably, and shows in-depth comprehension of the pertinent concepts and/or processes.

3

Poses a complex problem and finishes most significant parts of the solution, communicates ideas unmistakably, and shows comprehension of major concepts although neglects or misinterprets less significant ideas or details.

2 Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension.

1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach.

Source: D.O. #73, s. 2012 IV. LEARNING GOALS AND TARGETS

After going through this module, you should be able to demonstrate understanding of key concepts of circles and formulate real-life problems involving these concepts, and solve these using a variety of strategies. Furthermore, you should be able to investigate mathematical relationships in various situations involving circles.


DEPED COPY

135

J

A N

E

L

s

Start Lesson 1A of this module by assessing your knowledge of the

different mathematical concepts previously studied and your skills in performing mathematical operations. These knowledge and skills will help you understand circles. As you go through this lesson, think of this important question: “How do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions?” To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier. You may check your work with your teacher.

Use the figure below to identify and name the following terms related to A. Then, answer the questions that follow.

1. a radius 5. a minor arc

2. a diameter 6. a major arc 3. a chord 7. 2 central angles 4. a semicircle 8. 2 inscribed angles

Activity 1:


DEPED COPY

136

Questions: a. How did you identify and name the radius, diameter, and chord?

How about the semicircle, minor arc, and major arc? inscribed angle and central angle?

b. How do you describe a radius, diameter, and chord of a circle? How about the semicircle, minor arc, and major arc? inscribed angle and central angle?

Write your answers in the table below.

Terms Related to Circles Description

1. radius

2. diameter

3. chord

4. semicircle

5. minor arc

6. major arc

7. central angle

8. inscribed angle

c. How do you differentiate among the radius, diameter, and chord of a

circle? How about the semicircle, minor arc, and major arc? inscribed angle and central angle?

Were you able to identify and describe the terms related to circles? Were you able to recall and differentiate them? Now that you know the important terms related to circles, let us deepen your understanding of finding the lengths of sides of right triangles. You need this mathematical skill in finding the relationships among chords, arcs, and central angles as you go through this lesson.


DEPED COPY

137

In each triangle below, the length of one side is unknown. Determine the length of this side. 1. 4.

2. 5.

3. 6.

Questions:

a. How did you find the missing side of each right triangle? b. What mathematics concepts or principles did you apply to find each

missing side?

In the activity you have just done, were you able to find the missing

side of a right triangle? The concept used will help you as you go on with this module.

Activity 2:

a = 6

b = 8

c = ?

b = ?

a = 3

c = 5

a = 9

b = 9

c = ?

a = ?

b = 16

c = 20

a = 7

b = ?

c = 14

a = 9

b = 15

c = ?


DEPED COPY

138

O

Use the figures below to answer the questions that follow.

Figure 1 Figure 2

1. What is the measure of each of the following angles in Figure 1? Use a protractor. a. TOP d. ROS b. POQ e. SOT c. QOR

2. In Figure 2, AF , AB , AC , AD , and AE are radii of A. What is the

measure of each of the following angles? Use a protractor. a. FAB d. EAD b. BAC e. EAF c. CAD

3. How do you describe the angles in each figure? 4. What is the sum of the measures of ,TOP POQ , ,QOR ,ROS

and SOT in Figure 1?

How about the sum of the measures of ,FAB ,BAC ,CAD ,EAD and EAF in Figure 2?

Activity 3:

O T

S R

Q

P

C

B

F

D

E

A O


DEPED COPY

139

5. In Figure 1, what is the sum of the measures of the angles formed by the coplanar rays with a common vertex but with no common interior points?

6. In Figure 2, what is the sum of the measures of the angles formed by the radii of a circle with no common interior points?

7. In Figure 2, what is the intercepted arc of ?FAB How about ?BAC

CAD ? ?EAD ?EAF Complete the table below.

Central Angle Measure Intercepted Arc a. FAB b. BAC c. CAD d. EAD e. EAF

8. What do you think is the sum of the measures of the intercepted arcs

of ,FAB BAC , CAD , ,EAD and EAF ? Why? 9. What can you say about the sum of the measures of the central angles

and the sum of the measures of their corresponding intercepted arcs?

Were you able to measure the angles accurately and find the sum of their measures? Were you able to determine the relationship between the measures of the central angle and its intercepted arc? For sure you were able to do it. In the next activity, you will find out how circles are illustrated in real-life situations.

Use the situation below to answer the questions that follow. Rowel is designing a mag wheel like the one shown below. He decided to put 6 spokes which divide the rim into 6 equal parts.

Activity 4:


DEPED COPY

140

In the figure on the right, BAC is a central angle. Its sides divide A into arcs. One arc is the curve containing points B and C. The other arc is the curve containing points B, D, and C.

Recall that a central angle of a circle is an angle formed by two rays whose vertex is the center of the circle. Each ray intersects the circle at a point, dividing it into arcs.

Questions: a. What is the degree measure of each arc along the rim?

How about each angle formed by the spokes at the hub?

b. If you were to design a wheel, how many spokes will you use to divide the rim? Why?

How did you find the preceding activities? Are you ready to learn

about the relations among chords, arcs, and central angles of a circle? I am sure you are!!! From the activities done, you were able to recall and describe the terms related to circles. You were able to find out how circles are illustrated in real-life situations. But how do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions? You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on this lesson and the examples presented.

Central Angle and Arcs

Definition: Sum of Central Angles

(Note: All measures of angles and arcs are in degrees.)

The sum of the measures of the central angles of a circle with no common interior points is 360 degrees.

In the figure, 3604m3m2m1m .

4 3 2 1

B A

C

D

central angle

arc

arc


DEPED COPY

141

M

O

G E

Arcs of a Circle An arc is a part of a circle. The symbol for arc is . A semicircle is

an arc with a measure equal to one-half the circumference of a circle. It is named by using the two endpoints and another point on the arc.

Degree Measure of an Arc

1. The degree measure of a minor arc is the measure of the

central angle which intercepts the arc.

Example: GEO is a central angle. It intercepts E at

points G and O. The measure of GO is equal

to the measure of .GEO If m 118,GEO

then mGO = 118.

2. The degree measure of a major arc is equal to 360 minus the measure of the minor arc with the same endpoints.

Example: If mGO = 118, then mOMG = 360 – mGO.

That is, mOMG = 360 – 118 = 242.

Answer: mOMG = 242

3. The degree measure of a semicircle is 180 .

U

A minor arc is an arc of the circle that measures less than a semicircle. It is named usually by using the two endpoints of the arc.

Examples: JN, NE, and JE

A major arc is an arc of a circle that measures

greater than a semicircle. It is named by using the two endpoints and another point on the arc. Examples: JEN, JNE, and EJN

N

J

E

A

Z

O

C

N

Example: The curve from point N to point Z is an arc. It is part of O and is named as arc NZ or NZ. Other arcs of O are CN, CZ, CZN, CNZ, and NCZ.

If mCNZ is one-half the circumference

of O, then it is a semicircle.


DEPED COPY

142

W

T

S

E

65°

65°

N

M

K

I

M

A

T

H

Congruent Circles and Congruent Arcs

Congruent circles are circles with congruent radii.

Example: MA is a radius of A.

TH is a radius of T.

If THMA , then A T.

Congruent arcs are arcs of the same circle or of congruent circles with equal measures.

Example: In I, KSTM .

If I E, then NWTM and NWKS .

Theorems on Central Angles, Arcs, and Chords 1. In a circle or in congruent circles, two minor arcs are congruent if and only

if their corresponding central angles are congruent.

In E below, NEOSET Since the two central angles are

congruent, the minor arcs they intercept are also congruent. Hence,

NOST .

If E I and BIGNEOSET , then BGNOST .

65°

T

50° 50°

T

S N

O

E I

G

B

50°

.


DEPED COPY

143

Proof of the Theorem

The proof has two parts. Part 1. Given are two congruent circles and a central angle from each circle which are congruent. The two-column proof below shows that their corresponding intercepted arcs are congruent. Given: E I BIGSET

Prove: BGST Proof:

Statements Reasons

1. E I

BIGSET

1. Given

2. In E , .m SET mST

In I , .m BIG mBG

2. The degree measure of a minor arc is the measure of the central angle which intercepts the arc.

3. BIGmSETm

3. From 1, definition of congruent angles

4. mBGmST

4. From 2 & 3, substitution

5. BGST

5. From 4, definition of congruent arcs

Part 2. Given are two congruent circles and intercepted arcs from each

circle which are congruent. The two-column proof on the next page shows that their corresponding angles are congruent. Given: E I BGST

Prove: BIGSET

G B

I

E S

T

G B

I

E S

T


DEPED COPY

144

B

A

H

E O

N

C

T

Proof: Statements Reasons

1. E I

BGST

1. Given

2. In ,E .mST m SET

In I , .mBG m BIG


3. mBGmST

3. From 1, definition of congruent arcs

4. BIGmSETm

4. From 2 & 3, substitution

5. BIGSET

5. From 4, definition of congruent angles

Combining parts 1 and 2, the theorem is proven. 2. In a circle or in congruent circles, two minor arcs are

congruent if and only if their corresponding chords are congruent.

In T on the right, CHBA . Since

the two chords are congruent, then CHBA .

If T N and OECHBA , then

OECHBA .


DEPED COPY

145

B

A

T E

N

O

B

A

T E

N

O

Proof of the Theorem

The proof has two parts. Part 1. Given two congruent circles T N and two congruent corresponding chords AB and OE , the two-

column proof below shows that the corresponding minor arcs AB and OE are congruent. Given: T N OEAB Prove: OEAB Proof:

Statements Reasons 1. T N

OEAB 1. Given

2. NENOTBTA 2. Radii of the same circle or of

congruent circles are congruent.

3. ONEATB 3. SSS Postulate

4. ONEATB

4. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

5. OEAB

5. From the previous theorem, “In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.”

Part 2. Given two congruent circles T and N and two congruent minor arcs AB and OE , the two-column proof on the next page shows that the corresponding chords AB and OE are congruent. Given: T N OEAB Prove: OEAB


DEPED COPY

146

Proof: Statements Reasons

1. T N

OEAB 1. Given

2. mOEmAB

2. Definition of congruent arcs

3. BTA and ONE are central angles.

3. Definition of central angles

4. mBABTAm mOEONEm


5. ONEmBTAm

5. From 2, 4, substitution

6. NENOTBTA 6. Radii of the same circle or of

congruent circles are congruent.

7. ONEATB 7. SAS Postulate

8. OEAB 8. Corresponding Parts of Congruent

Triangles are Congruent (CPCTC)

Combining parts 1 and 2, the theorem is proven. 3. In a circle, a diameter bisects a chord and an arc with the same endpoints

if and only if it is perpendicular to the chord.

The proof of the theorem is given as an exercise in Activity 9.

N

E

G

U I S

In U on the right, ES is a diameter and

GN is a chord. If GNES , then INGI and

ENGE .


DEPED COPY

147

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Example: Adjacent arcs are arcs with exactly one point in common. In E, LO and OV

are adjacent arcs. The sum of their measures is equal to the measure of

LOV. If mLO = 71 and mOV = 84, then mLOV = 71 + 84 = 155. Sector and Segment of a Circle A sector of a circle is the region bounded by an arc of the circle and the two radii to the endpoints of the arc. To find the area of a sector of a

circle, get the product of the ratio 360

arctheofmeasure and the area of the

circle. Example: The radius of C is 10 cm. If mAB = 60, what is the

area of sector ACB? Solution: To find the area of sector ACB:

a. Determine first the ratio

360

arctheofmeasure.

360

60

360

arctheofmeasure

6

1

b. Find the area (A) of the circle using the equation A = 2r ,

where r is the length of the radius.

A = 2r

= 2cm10

= 2cm100

O E

V

L

C B

A

10 cm

60°


DEPED COPY

148

c. Get the product of the ratio 360

arctheofmeasure and the

area of the circle.

Area of sector ACB = 2cm1006

1

= 2cm3

50

The area of sector ACB is 2cm3

50.

A segment of a circle is the region bounded by an arc and the segment joining its endpoints.

Example: The shaded region in the figure below is a segment of

T. It is the region bounded by PQ and PQ .

In the same figure, the area of ΔPTQ = cm5cm5

2

1 or

ΔPTQ = 2cm2

25.

The area of the shaded segment, then, is equal to 2cm2

25

4

25

which is approximately 7.135 cm2.

To find the area of the shaded segment in the figure, subtract the area of triangle PTQ from the area of sector PTQ.

If mPQ = 90 and the radius of the circle

is 5 cm, then the area of sector PTQ is one-fourth of the area of the whole circle. That is,

Area of sector PTQ =

25

4

1cm

=

2cm254

1

= 2cm4

25


DEPED COPY

149

Arc Length

The length of an arc can be determined by using the proportion

360 2

Ar

l , where A is the degree measure of the arc, r is the radius of the

circle, and l is the arc length. In the given proportion, 360 is the degree measure of the whole circle, while 2r is the circumference. Example: An arc of a circle measures 45°. If the radius of the circle is 6 cm,

what is the length of the arc?

Solution: In the given problem, A = 45 and r = 6 cm. To find l, the equation

360 2

Ar

l can be used. Substitute the given values in the

equation.

360 2

Ar

l 45360 2 (6)

l 18 12

l

128 l 4.71l

The length of the arc is approximately 4.71 cm.

Learn more about Chords, Arcs, Central Angles, Sector, and Segment of a Circle through the WEB. You may open the following links.

http://www.cliffsnotes.com/math/geometry/ circles/central-angles-and-arcs http://www.mathopenref.com/arc. html http://www.mathopenref.com/chord.html http://www.mathopenref.com/circlecentral. html http://www.mathopenref.com/arclength.html http://www.mathopenref.com/arcsector.html http://www.mathopenref.com/segment.html http://www.math-worksheet.org/arc-length-and-sector-area


http://www.cliffsnotes.com/math/geometry/%20circles/central-angles-and-arcs

http://www.cliffsnotes.com/math/geometry/%20circles/central-angles-and-arcs

http://www.mathopenref.com/arc.html

http://www.mathopenref.com/chord.html

http://www.mathopenref.com/circlecentral.%20html

http://www.mathopenref.com/arclength.html

http://www.mathopenref.com/arcsector.html

http://www.math-worksheet.org/arc-length-and-sector-area


DEPED COPY

150

Your goal in this section is to apply the key concepts of chords,

arcs, and central angles of a circle. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided.

Use A below to identify and name the following. Then, answer the questions that follow.

1. 2 semicircles in the figure

2. 4 minor arcs and their corresponding major arcs

3. 4 central angles

Questions:

a. How did you identify and name the semicircles? How about the minor arcs and the major arcs? central angles?

b. Do you think the circle has more semicircles, arcs, and central angles? Show.

Were you able to identify and name the arcs and central angles in

the given circle? In the next activity, you will apply the theorems on arcs and central angles that you have learned.

Activity 5:

H

K L

M

G

A

J


DEPED COPY

151

In A below, m 42,LAM m 30,HAG and KAH is a right angle. Find the following measure of an angle or an arc, and explain how you arrived at your answer.

1. LAKm 6. LKm 2. JAKm 7. JKm 3. LAJm 8. LMGm 4. JAHm 9. JHm 5. KAMm 10. KLMm

In the activity you have just done, were you able to find the degree measure of the central angles and arcs? I am sure you did! In the next activity, you will apply the relationship among the chords, arcs, and central angles of a circle.

In the figure, JI and ON are diameters of S. Use the figure and the given information to answer the following.

1. Which central angles are congruent? Why?

2. If 113m JSN , find: a. ISOm

b. NSIm

c. JSOm

3. Is INOJ ? How about JN and OI ? Justify your answer.

4. Which minor arcs are congruent? Explain your answer.

5. If 67m JSO , find:

a. JOm d. IOm

b. JNm e. JOmN

c. NIm f. NIOm 6. Which arcs are semicircles? Why?

Activity 7:

Activity 6:

I

J

S

N

O

H

K L

M

G

A

J


DEPED COPY

152

Were you able to apply the relationship among the chords, arcs and

central angles of a circle? In Activity 8, you will use the theorems on chords in finding the lengths of chords.

In M below, BD = 3, KM = 6, and KP = 72 . Use the figure and the given information to find each measure. Explain how you arrived at your answer.

1. AM 5. DS

2. KL 6. MP 3. MD 7. AK 4. CD 8. KP

Were you able to find the length of the segments? In the next activity, you will complete the proof of a theorem on central angles, arcs, and chords of a circle.

Complete the proof of the following theorem.

In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is perpendicular to the chord. Given: ES is a diameter of U and perpendicular to chord GN at I. Prove: 1. GINI 2. EGEN 3. GSNS

Activity 9:

Activity 8:

D

M

3

C

6

72

A

S

P

L

B

K

N G

E

I

S

U


DEPED COPY

153

Proof of Part 1: Show that ES bisects GN and the minor arc GN.

Statements Reasons 1. U with diameter ES and chord

GN GNES

Two points determine a line.

2. GIU and NIU are right angles. Given

3. Lines that are perpendicular tm right angles.

4. UNUG Radii of a circle are congruent.

5. UIUI Reflexive Property of Congruence.

6. NIUGIU

7. NIGI Corresponding parts of congruent triangles are congruent.

8. ES bisects GN .

Corresponding parts of congruent triangles are congruent

9. NUIGUI

In a circle, congruent central angles intercept congruent arcs.

10. GUI and GUE are the same

angles. NUI and NUE are the same

angles.

Two angles that form a linear pair are supplementary.

11. NUEmGUEm Supplements of congruent angles are congruent.

12. GUEmmEG

NUEmmEN

In a circle, congruent central angles intercept congruent arcs.

13. mEGmEN

14. NUSmGUSm

15. GUSmmGS

NUSmmNS

16. mGSmNS

17. ES bisects GN .

;

GIU NIU


DEPED COPY

154

Given: ES is a diameter of U; ES bisects GN at I and the minor arc GN.

Proof of Part 2: Show that GNES .

Statements Reasons 1. U with diameter ES , ES bisects

GN at I and the minor arc

GN.

Two points determine a line.

2. NIGI

NEGE

Given

3.

4. UNUG Radii of a circle are congruent.

5. NIUGIU Reflexive Property of Congruence.

6. UINUIG

7. UIG and UIN are right angles.

Corresponding parts of congruent triangles are congruent.

8. GNIU

9. GNES

Was the activity interesting? Were you able to complete the proof? You will do more of this in the succeeding lessons. Now, use the ideas you have learned in this lesson to find the arc length of a circle.

S

N G I

U

E

UI UI

S


DEPED COPY

155

The radius of O below is 5 units. Find the length of each of the following arcs given the degree measure. Answer the questions that follow.

1. mPV = 45; length of PV = ________

2. mPQ = 60; length of PQ = ________

3. mQR = 90; length of QR = ________

4. mRTS = 120; length of RTS = ________

5. mQRT = 95; length of QRT = ________

Questions:

a. How did you find the length of each arc? b. What mathematics concepts or principles did you apply to find the

length of each arc?

Were you able to find the arc length of each circle? Now, find the area of the shaded region of each circle. Use the knowledge learned about segment and sector of a circle in finding each area.

Find the area of the shaded region of each circle. Answer the questions that follow. 1. 2. 3.

Activity 11:

Activity 10:

B C

A

R

Q

S

X

Z Y 6 cm

90°

12 cm

45°

8 cm

135°

T

S

P Q

R

O r = 5 V


DEPED COPY

156

4. 5. 6.

Questions:

a. How did you find the area of each shaded region? b. What mathematics concepts or principles did you apply to find the area

of the shaded region? Explain how you applied these concepts.

How was the activity you have just done? Was it easy for you to find the area of segments and sectors of circles? It was easy for sure!

In this section, the discussion was about the relationship among chords, arcs, and central angles of circles, arc length, segment and sector of a circle, and the application of these concepts in solving problems.

Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion?

Now that you know the important ideas about this topic, let us go deeper by moving on to the next section.

T X

S

4 cm

A R

E

B

W M

5 cm

100°

J

S

O

6 cm

Y


DEPED COPY

157

Your goal in this section is to take a closer look at some aspects of the

topic. You are going to think deeper and test further your understanding of circles. After doing the following activities, you should be able to answer this important question: “How do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions?”

Answer the following questions. 1. Five points on a circle separate the circle into five congruent arcs.

a. What is the degree measure of each arc?

b. If the radius of the circle is 3 cm, what is the length of each arc?

c. Suppose the points are connected consecutively with line segments. How do you describe the figure formed?

2. Do you agree that if two lines intersect at the center of a circle, then the lines

intercept two pairs of congruent arcs? Explain your answer.

3. In the two concentric circles on the right,

CON intercepts CN and RW. a. Are the degree measures of CN and RW

equal? Why?

b. Are the lengths of the two arcs equal? Explain your answer.

4. The length of an arc of a circle is 6.28 cm. If the circumference of the circle is

37.68 cm, what is the degree measure of the arc? Explain how you arrived at your answer.

5.

Activity 12:

O

R

W N

C

Mr. Lopez would like to place a fountain in his circular garden on the right. He wants the pipe, where the water will pass through, to be located at the center of the garden. Mr. Lopez does not know where it is. Suppose you were asked by Mr. Lopez to find the center of the garden, how would you do it?


DEPED COPY

158

6. The monthly income of the Soriano family is Php36,000.00. They spend Php9,000.00 for food, Php12,000.00 for education, Php4,500.00 for utilities, and Php6,000.00 for other expenses. The remaining amount is for their savings. This information is shown in the circle graph below. a. Which item is allotted with the highest budget? How about the least?

Explain.

b. If you were to budget your family’s monthly income, which item would you give the greater allocation? Why?

c. In the circle graph, what is the measure of the central angle

corresponding to each item?

d. How is the measure of the central angle corresponding to each item determined?

e. Suppose the radius of the circle graph is 25 cm. What is the area of

each sector in the circle graph? How about the length of the arc of each sector?

In this section, the discussion was about your understanding of chords, arcs, central angles, area of a segment and a sector, and arc length of a circle including their real-life applications.

What new realizations do you have about the lesson? How would you connect this to real life?

Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section.

Soriano Family’s Monthly Expenses


DEPED COPY

159

Your goal in this section is to apply your learning to real-life

situations. You will be given a practical task which will demonstrate your understanding of circles.

Answer the following. Use the rubric provided to rate your work. 1. Name 5 objects or cite 5 situations in real life where chords, arcs, and

central angles of a circle are illustrated. Formulate problems out of these objects or situations, then solve.

2. Make a circle graph showing the different school fees that students like you have to pay voluntarily. Ask your school cashier how much you would pay for the following school fees: Parents-Teachers Association, miscellaneous, school paper, Supreme Student Government, and other fees. Explain how you applied your knowledge of central angles and arcs of a circle in preparing the graph.

3. Using the circle graph that you made in number 2, formulate at least two

problems involving arcs, central angles, and sectors of a circle, then solve. Rubric for a Circle Graph

Score Descriptors 4 The circle graph is accurately made, presentable, and appropriate.

3 The circle graph is accurately made and appropriate but not presentable.

2 The circle graph is not accurately made but appropriate. 1 The circle graph is not accurately made and not appropriate.

Activity 13:


DEPED COPY

160


Score Descriptors

6

Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate.

5 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

3

Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details.



Source: D.O. #73, s. 2012

In this section, your task was to name 5 objects or cite 5 situations in real life where chords, arcs, and central angles of a circle are illustrated. Then, formulate and solve problems out of these objects or situations. You were also asked to make a circle graph.

How did you find the performance task? How did the task help you realize the importance of the lesson in real life?

SUMMARY/SYNTHESIS/GENERALIZATION: This lesson was about the relationships among chords, arcs, and central

angles of a circle, area of a segment and a sector, and arc length of a circle. In this lesson, you were asked to determine the relationship between the measures of the central angle and its intercepted arc. You were also given the opportunity to apply the different geometric relationships among chords, arcs, and central angles in solving problems, complete the proof of a theorem related to these concepts, find the area of a segment and the sector of a circle, and determine the length of an arc.

Moreover, you were asked to name objects and cite real-life situations

where chords, arcs, and central angles of a circle are illustrated and applied. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the next lesson, Arcs and Inscribed Angles of Circles.


DEPED COPY

161

Start Lesson 1B of this module by checking your prior mathematical

knowledge and skills that will help you in understanding the relationships among arcs and inscribed angles of a circle. As you go through this lesson, think of this important question: How are the relationships among arcs and inscribed angles of a circle used in finding solutions to real-life problems and in making decisions? To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier. You may check your work with your teacher’s guidance.

Name the angles and their intercepted arcs in the figure below. Answer the questions that follow.

Angles Arc That the Angle Intercepts

S

D

C M

G

Activity 1:


DEPED COPY

162

Questions:

1. How did you identify and name the angles in the figure? How about the arcs that these angles intercept?

2. How many angles did you identify in the figure?

How about the arcs that these angles intercept?

3. When do you say that an angle intercepts an arc?

Were you able to identify and name the angles and their intercepted arcs? I am sure you were! This time, find out the relationships that exist among arcs and inscribed angles of a circle by doing the next activity.

Perform the following activity by group. Answer every question that follows. Procedure: 1. Use a compass to draw a circle. Mark and label the center of the circle as

point E. 2. Draw a diameter of the circle. Label the endpoints as D and W. 3. From the center of the circle, draw radius EL.

Using a protractor, what is the measure of ?LEW

How about the degree measure of LW? Why? 4. Draw LDW by connecting L and D with a line segment.

Using a protractor, what is the measure of ?LDW 5. LDW is an inscribed angle.

How do you describe an inscribed angle? 6. LW is the intercepted arc of .LDW Compare the measure of LDW

with the degree measure of LW. What statements can you make?

Activity 2:


DEPED COPY

163

7. Draw other inscribed angles in the circle. Determine the measures of these angles and the degree measures of their respective intercepted arcs.

How does the measure of each inscribed angle compare with the degree measure of its intercepted arc?

What conclusion can you make about the relationship between the measure of an inscribed angle and the measure of its intercepted arc?

Were you able to determine the relationship between the measure

of an inscribed angle and the measure of its intercepted arc? If yes, then you are now ready to determine the relationship that exists when an inscribed angle intercepts a semicircle by performing the next activity.

Perform the following activity by group. Answer every question that follows. Procedure:

1. Draw a circle whose radius is 3 cm. Mark the center and label it C. 2. Extend the radius to form a diameter of 6 cm. Mark and label the

endpoints of the diameter with M and T. 3. On the semicircle, mark and label three points O, U, and N. 4. Draw three different angles whose vertices are O, U, and N, respectively,

and whose sides contain M and T. 5. Find the measure of each of the following angles using a protractor.

a. MOT b. MUT c. MNT

What can you say about the measures of the angles?

What statements can you make about an inscribed angle intercepting a semicircle?

How would you compare the measures of inscribed angles intercepting the same arc?

Activity 3:


DEPED COPY

164

Were you able to determine the measure of an inscribed angle that intercepts a semicircle? For sure you were able to do it. In the next activity, you will find out how inscribed angles are illustrated in real-life situations.

Use the situation below to answer the questions that follow.

Janel works for a realtor. One of her jobs is to take photographs of

houses that are for sale. She took a photograph of a house two months ago using a camera lens with 80° field of view like the one shown below. She has returned to the house to update the photo, but she has forgotten her lens. Now, she only has a telephoto lens with a 40° field of view.

Questions:

1. From what location(s) could Janel take a photograph of the house with the telephoto lens, so that the entire house still fills the width of the picture? Use an illustration to show your answer.

2. What mathematics concept would you apply to show the exact location of the photographer?

3. If you were the photographer, what would you do to make sure that the entire house is captured by the camera?

Activity 4:


DEPED COPY

165

Figure 2

T

O

P

How did you find the preceding activities? Are you ready to learn about the relations among arcs and inscribed angles of a circle? I am sure you are! From the activity done, you were able to find out how inscribed angles are used in real-life situations. But how does the concept of inscribed angles of a circle facilitate finding solutions to real-life problems and making decisions? You will find these out through the activities in the next section. Before doing these activities, read and understand first some important notes on this lesson and the examples presented.

Inscribed Angles and Intercepted Arcs

An inscribed angle is an angle whose vertex is on a circle and whose

sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

Examples: Figure 1

In Figure 1, LAP is an inscribed angle and its intercepted arc is LP.

The center of the circle is in the interior of the angle.

In Figure 2, TOP is an inscribed angle and its intercepted arc is TP.

One side of the angle is the diameter of the circle.

In Figure 3, CGM is an inscribed angle and its intercepted arc is CM.

The center of the circle is in the exterior of the angle. Theorems on Inscribed Angles

1. If an angle is inscribed in a circle, then the measure of the angle equals

one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). Note: The theorem has three cases and the proof of each case is given as an exercise in Activity 8 and Activity 9.

Example: In the figure on the right, ACT is

an inscribed angle and AT is its intercepted arc.

If mAT = 120, then ACTm = 60.

P

A

L

G

Figure 3

M

C

T

A

C


DEPED COPY

166

N

T O

S

E

2. If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent.

Example 1: In Figure 1 below, PIO and PLO intercept PO. Since

PIO and PLO intercept the same arc, the two angles, then, are congruent.

Example 2: In Figure 2 above, SIM and ELP intercept SM and EP, respectively. If EPSM , then ELPSIM .

3. If an inscribed angle of a circle intercepts a semicircle, then the angle is a

right angle.

Example: In the figure, NTE intercepts NSE. If NSE is a semicircle, then NTE is a right angle.

4. If a quadrilateral is inscribed in a circle, then its opposite angles are

supplementary.

Example: Quadrilateral DREA is inscribed in M. 180 REAmRDAm . 180 DAEmDRAm .

M

L I

E

P

S

Figure 2 P O

I

T

L

Figure 1

E A

R

D M


DEPED COPY

167

N

Learn more about Arcs and Inscribed Angles of a Circle through the WEB. You may open the following links.

http://www.cliffsnotes.com/math/geometry/circles/arcs-and-inscribed-angles

http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/section/8.4/

http://www.math-worksheet.org/inscribed-angles

http://www.mathopenref.com/circleinscribed.html

http://www.onlinemathlearning.com/circle-theorems.html

Your goal in this section is to apply the key concepts of arcs and

inscribed angles of a circle. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided.

In the figure below, CE and LA are diameters of N. Use the figure to answer the following. 1. Name all inscribed angles in the figure.

2. Which inscribed angles intercept the following arcs?

a. CL c. LE

b. AE d. AC

Activity 5:

8

6

A

C

L

E

2 3

1

9

4

7

5




DEPED COPY

168

3. If mLE = 124, what is the measure of each of the following angles?

a. 1 d. 4 g. 7

b. 2 e. 5 h. 8

c. 3 f. 6 i. 9

4. If 1 26,m what is the measure of each of the following arcs?

a. CL c. AE

b. AC d. LE

Were you able to identify the inscribed angles and their intercepted arcs including their degree measures? In the next activity, you will apply the theorems on arcs and inscribed angles that you have learned.

In F, AB , BC , CD , BD and AC are chords. Use the figure and the given information to answer the following questions. 1. Which inscribed angles are congruent?

Explain your answer.

2. If 54CBDm , what is the measure of CD?

3. If 96mAB , what is the measure of ACB ? 4. If 35 xABDm and 104 xDCAm , find:

a. the value of x c. DCAm b. ABDm d. mAD

5. If 46 xBDCm and 210 xmBC , find:

a. the value of x c. mBC b. BDCm d. BACm

In the activity you have just done, were you able to apply the theorems on arcs and inscribed angles? I am sure you were! In the next activity, you will still apply the theorems you have studied in this lesson.

Activity 6:

A

D

E

F

C

B


DEPED COPY

169

Use the given figures to answer the following.

1. ∆GOA is inscribed in L. If 75m OGA and 160m AG , find:

a. OAm c. GOAm

b. OGm d. GAOm 2. Isosceles ∆CAR is inscribed in E. If m 130,CR find:

a. CARm

b. ACRm

c. ARCm

d. ACm

e. ARm

3. DR is a diameter of O. If 70m MR , find: a. RDMm d. DMm

b. DRMm e. RDm

c. DMRm

4. Quadrilateral FAIT is inscribed in H. If 75m AFT and 98m FTI , find: a. TIAm

b. FAIm

Activity 7:

A O

G

75°

L

160°

A

C

R 130°

E

D

O

M 70° R

F

A

I

75°

T 98°

H


DEPED COPY

170

5. Rectangle TEAM is inscribed in B. If 64m TE and 58m TEM , find: a. TMm

b. MAm

c. AEm

d. MEAm

e. TAMm

How was the activity you have just done? Was it easy for you to apply the theorems on arcs and inscribed angles? It was easy for sure!

Now, let us complete the proof of a theorem on inscribed angle and its intercepted arc.

Complete the proof of the theorem on inscribed angle and its intercepted arc.

The proofs of cases 2 and 3 of this theorem are given in Activity 9.

If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). Case 1: Given: PQR inscribed in S and PQ is a diameter.

Prove: PRPQR m21m

Draw RS and let xPQR m .

Activity 8:

P R

Q

S

x

T

B

E

M

A


DEPED COPY

171

Statements Reasons 1. PQR is inscribed in S and

PQ is a diameter.

2. RSQS 3. QRS is an isosceles .

4. QRSPQR

5. QRSPQR mm

6. xQRS m

7. xPSR 2m

8. PRPSR mm

9. xPR 2m

10. PQRPR m2m

11. PRQRS m2

1m

Were you able to complete the proof of the first case of the theorem? I

know you did!

In this section, the discussion was about the relationship among arcs and inscribed angles of a circle.

Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Which ideas are different and need modification?

Now that you know the important ideas about this topic, let us go

deeper by moving on to the next section.


DEPED COPY

172

Your goal in this section is to take a closer look at some aspects of the topic. You are going to think deeper and test further your understanding of the relationships among inscribed angles and their intercepted arcs. After doing the following activities, you should be able to answer this important question: How are the relationships among inscribed angles and their intercepted arcs applied in real-life situations?

Write a proof of each of the following theorems. 1. If an angle is inscribed in a circle, then the measure of the angle equals

one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle).

Case 2:

Given: KLM inscribed in O.

Prove: KMKLM m21m

Case 3:

Given: SMC inscribed in A.

Prove: Cm21m SSMC

Activity 9:

S

C

M

A

K

M

L

O


DEPED COPY

173


Given: In T, PR and AC are the

intercepted arcs of PQR

and ABC , respectively.

ACPR

Prove: ABCPQR


right angle.

Given: In C, GML intercepts semicircle GEL.

Prove: GML is a right angle.

4. If a quadrilateral is inscribed in a circle, then its opposite angles are

supplementary.

Given: Quadrilateral WIND is inscribed in Y .

Prove: 1. W and N are supplementary. 2. I and D are supplementary.

E G

C

M

L

A

B

C P

R Q

T

W

I

D

N Y


DEPED COPY

174

Were you able to prove the theorems on inscribed angles and intercepted arcs? In the next activity, you will use these theorems to prove congruence of triangles.

Write a two-column proof for each of the following. 1. MT and AC are chords of D. If ATMC ,

prove that THACHM .

2. Quadrilateral DRIV is inscribed in E. RV is a diagonal that passes through the center of the circle. If IVDV , prove that . RVD RVI

3. In A, NESE and NTSC . Prove that TNECSE .

Were you able to use the theorems on inscribed angles to prove

congruence of triangles? In the next activity, you will further understand inscribed angles and how they are used in real life.

Activity 10:

M

A

T D

C

H

E I

V

R

D

S

C

E

N

T A


DEPED COPY

175

Answer the following questions. 1. There are circular gardens having paths in the

shape of an inscribed regular star like the one shown on the right.

2. What kind of parallelogram can be inscribed in a circle? Explain. 3. The chairs of a movie house are arranged consecutively like an arc of a

circle. Joanna, Clarissa, and Juliana entered the movie house but seated away from each other as shown below.

Let E and G be the ends of the screen and F be one of the seats. The angle formed by E, F, and G or EFG is called the viewing angle of the person seated at F. Suppose the viewing angle of Clarissa in the above figure measures 38°. What are the measures of the viewing angles of Joanna and Juliana? Explain your answer.

4. A carpenter’s square is an L-shaped tool used to draw right angles.

Mang Ador would like to make a copy of a circular plate using the available wood that he has. Suppose he traces the plate on a piece of wood. How could he use a carpenter’s square to find the center of the circle?

Activity 11:

a. Determine the measure of an arc intercepted by an inscribed angle formed by the star in the garden.

b. What is the measure of an inscribed angle in a garden with a five-pointed star? Explain.

Movie Screen

Joanna

Clarissa

Juliana

E G

380

F


DEPED COPY

176

5.

In this section, the discussion was about your understanding of

inscribed angles and how they are used in real life. What new realization do you have about inscribed angles? How would

you connect this to real life?


Your goal in this section is to apply your learning to real-life situations.

You will be given a practical task which will demonstrate your understanding of inscribed angles.

Make a design of a stage where a special event will be held. Include in the design some circular objects that illustrate the use of inscribed angles and arcs of a circle. Explain how you applied your knowledge of inscribed angles and intercepted arcs of a circle in preparing the design. Then, formulate and solve problems out of this design that you made. Use the rubric provided to rate your work.

Activity 12:

P Q

R

S

T

M

N

Ramon made a circular cutting board by sticking eight 1- by 2- by 10-inch boards together, as shown on the right. Then, he drew and cut a circle with an 8-inch diameter from the boards. a. In the figure, if PQ is a diameter of the

circular cutting board, what kind of triangle is PQR ?

b. How is RS related to PS and QS ? Justify your answer.

c. Find PS, QS, and RS. d. What is the length of the seam of the

cutting board that is labeled RT ? How about MN ?


DEPED COPY

177

Rubric for a Stage’s Design

Score Descriptors 4 The stage’s design is accurately made, presentable, and appropriate.

3 The stage’s design is accurately made and appropriate but not presentable.

2 The stage’s design is not accurately made but appropriate. 1 The stage’s design is not accurately made and not appropriate.


Score Descriptors

6

Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes and provides explanations wherever appropriate.



3




Source: D.O. #73, s. 2012

In this section, your task was to design a stage, formulate, and solve problems where inscribed angles of circles are illustrated.

How did you find the performance task? How did the task help you realize the importance of the topic in real life?

SUMMARY/SYNTHESIS/GENERALIZATION:

This lesson was about arcs and inscribed angles of a circle. In this lesson, you were given the opportunity to determine the geometric relationships that exist among arcs and inscribed angles of a circle, apply these in solving problems, and prove related theorems. Moreover, you were given the chance to formulate and solve real-life problems involving these geometric concepts. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the next lesson, Tangent and Secant Segments.


DEPED COPY

178

Start Lesson 2A of this module by assessing your knowledge of the

different mathematical concepts previously studied and other mathematical skills learned. These knowledge and skills will help you understand the different geometric relationships involving tangents and secants of a circle. As you go through this lesson, think of this important question: How do the different geometric relationships involving tangents and secants of a circle facilitate finding solutions to real-life problems and making wise decisions? To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier. You may check your work with your teacher.

Perform the following activity. Answer every question that follows. Procedure:

1. Use a compass to draw S.

2. Draw line m such that it intersects S at exactly one point. Label the point of intersection as T.

3. Connect S and T with a line segment. What is TS in the figure drawn?

4. Mark four other points on line m such that two of these points are on the left side of T and the other two points are on the right side. Label these points as M, N, P, and Q, respectively.

5. Using a protractor, find the measures of ,MTS ,NTS ,PTS and QTS . How do the measures of the four angles compare?

6. Repeat steps 2 to 5. This time, draw line n such that it intersects the circle at another point. What statement can you make about the measures of angles in item #5 and those in item #6?

7. Draw ,MS ,NS PS , and QS .

Activity 1:


DEPED COPY

179

8. Using a ruler, find the lengths of TS , MS , NS , PS , and QS . How do the lengths of the five segments compare?

What do you think is the shortest segment from the center of a circle to the line that intersects it at exactly one point? Explain your answer.

In the activity you have just done, were you able to compare the

measures of different angles drawn? Were you able to determine the shortest segment from the center of a circle to the line that intersects it at exactly one point? I know you were! The activity you have done has something to do with your new lesson. Do you know why? Find this out in the succeeding activities!

In the figure below, C is the center of the circle. Use the figure to answer the questions that follow.

1. Which lines intersect circle C at two points?

How about the lines that intersect the circle at exactly one point?

2. What are the angles having A as the vertex? C as the vertex? D as the vertex? G as the vertex? Make a list of these angles, then describe each.

3. What arc/s does each angle intercept?

Activity 2:


DEPED COPY

180

4. Which angles intercept the same arc? 5. Using a protractor, find the measures of the angles identified in item #2? 6. How would you determine the measures of the arcs intercepted by the

angles? Give the degree measure of each arc. 7. Compare the measures of DCE and DAE .

How about the mDE and m DAE ? Explain your answer. 8. How is the mAD related to the m DAB ? How about mEFA and m EAG ?

9. What relationship exists among mAD, mAF, and m BGD ?

Were you able to measure the different angles and arcs shown in the figure? Were you able to find out the different relationships among these angles and arcs? Learn more about these relationships in the succeeding activities.

Prepare the following materials, then perform the activity that follows. Answer every question asked.

Materials: Circular cardboard with radius 6 cm that is equally divided into

72 arcs so that each arc measures 5°

2 pieces of string, each measures about 40 cm

self-adhesive tape

cardboard or any flat surface Procedure:

Activity 3:

1. Attach the endpoints of the strings to the cardboard or any flat surface using self-adhesive tape to form an angle of any convenient measure. Label the angle as RST.

R

S T


DEPED COPY

181

2.

If the edge of the circular cardboard represents a circle, what is

RST in relation to the circle?

What are the measures of RST and RT? Explain how you arrived at your answer.

How would you compare the measure of RST with that of ST?

Is there any relationship among the measures of RST , RVT,

and RT? Describe the relationship, if there is any.

R S

T

2. Locate the center of the circular cardboard. Slide it underneath the strings until its center coincides with their point of intersection, S.

3. Slide the circular cardboard so that RS intersects the circle at S and ST intersects the circle at two points, S and T.

R

S T

4. Find the measure of ST using the circular cardboard.

R

S

T

V

5. Slide the circular cardboard so that S is

in the exterior of the circle and RS and

ST intersect the circle at R and T, respectively. Mark and label another point V on the circle.

6. Find the measures of RVT and RT.


in the exterior of the circle, ST intersects

the circle at T, and RS intersects the circle at two points, R and N.

R

S T

N


DEPED COPY

182

Is there any relationship among the measures of RST , RT, and NT? Describe the relationship, if there is any.

Is there any relationship among the measures of RST , RT, and MN? Describe the relationship, if there is any.

Is there any relationship among the measures of RST , RT, and MN? Describe the relationship, if there is any.

Was the activity interesting? Were you able to come up with some relationships involving angles formed by lines and their intercepted arcs? Are you ready to learn about tangents and secants and their real-life applications? I am sure you are! “How do the different geometric relationships involving tangents and secants of a circle facilitate finding solutions to real-life problems and making wise decisions?” You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on tangents and secants and the different geometric relationships involving them. Understand very well the examples presented so that you will be guided in doing the succeeding activities.


in the exterior of the circle, RS intersects the circle at points N and R,

and ST intersects the circle at points M and T.

8. Find the measures of RT and NT.

10. Find the measures of RT and MN.

R

S

T

N

M

11. Slide the circular cardboard so that S

is in the interior of the circle, NT intersects the circle at points N and T,

and MR intersects the circle at points M and R.

R

S T

N

M

12. Find the measures of RT and MN.


DEPED COPY

183

A

V

U

B

Q If AB is tangent to Q at R, then it is perpendicular to radius QR.

C

L

T

S

If CS is perpendicular to radius LT at L, then it is tangent to T.

Tangent Line

A tangent to a circle is a line coplanar with the circle and intersects it in one and only one point. The point of intersection of the line and the circle is called the point of tangency.

Example: Postulate on Tangent Line

At a given point on a circle, one and only one line can be drawn that is tangent to the circle.

Theorems on Tangent Line 1. If a line is tangent to a circle, then it is perpendicular

to the radius drawn to the point of tangency.

2. If a line is perpendicular to a radius of a circle at its endpoint that is on the

circle, then the line is tangent to the circle.

To illustrate, consider V on the right. If U is a point on the circle, then one and only one line can be drawn through U that is tangent to the circle.

R

A

In the figure on the right, PQ

intersects C at A. PQ is a

tangent line and A is the point of

tangency.

C Q

A

P


DEPED COPY

184

3. If two segments from the same exterior point are tangent to a circle, then the two segments are congruent.

Common Tangent

A common tangent is a line that is tangent to two circles in the same plane.

Common internal tangents intersect the segment joining the centers of the two circles

Common external tangents do not intersect the segment joining the centers of the two circles.

Tangent and Secant

Segments and rays that are contained in the tangent or intersect the circle in one and only one point are also said to be tangent to the circle.

Lines s and t are common external tangents.

tangents.

c

D E

d

Lines c and d are common internal tangents.

tangents.

M N

t

sn

If DW and GW are tangent to E, then GWDW .

D

E

G W

M N

Q

R

S

In the figure on the right, MN

and QR are tangent to S.


DEPED COPY

185

A

M

N

A secant is a line that intersects a circle at exactly two points. A secant contains a chord of a circle. Theorems on Angles Formed by Tangents and Secants 1. If two secants intersect in the exterior of a circle, then the measure of the

angle formed is one-half the positive difference of the measures of the intercepted arcs. In the figure below, NX and MY are two secants intersecting outside the circle at point P. XY and MN are the two intercepted arcs of .XPY

2. If a secant and a tangent intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

P

X

Y

N

M

MNXYXPY mm2

1m

For example, if mXY = 140 and mMN = 30, then

301402

1m XPY

1102

1

55m XPY

In circle A, MN is a secant line.


DEPED COPY

186

In the figure below, CM is a secant and LM is a tangent intersecting

outside the circle at point M. LEC and LG are the two intercepted arcs of

LMC .

3. If two tangents intersect in the exterior of a circle, then the measure of the

angle formed is one-half the positive difference of the measures of the intercepted arcs.

In the figure below, QK and QH are two tangents intersecting outside

the circle at point Q. HJK and HK are the two intercepted arcs of KQH .

4. If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

H Q

K

J

HKHJKKQH mm2

1m

For example, if mHJK = 250 and mHK = 110, then

1102502

1m KQH

1402

1

70m KQH

E

C

L M

G

LGLECLMC mm2

1m

For example, if mLEC = 186 and mLG = 70, then

701862

1m LMC

1162

1

58m LMC


DEPED COPY

187

S T

R

Q

W

In the figure below, WS and RX are two secants intersecting inside the

circle. WR and XS are the two intercepted arcs of 1 while WX and RS

are the two intercepted arcs of 2 .

5. If a secant and a tangent intersect at the point of tangency, then the

measure of each angle formed is one-half the measure of its intercepted arc.

In the figure below, QS is a secant and RW is a tangent intersecting at

S, the point of tangency. QS is the intercepted arc of QSR while QTS is

the intercepted arc of QSW .

XSWR mm2

11m

For example,

if mWR = 100 and

mXS = 120, then

1101m

2202

1

1201002

11m

Smm2

12m RWX

For example,

if mWX = 80 and

mRS = 60, then

702m

1402

1

60802

12m

QSQSR m2

1m

For example,

if mQS = 170, then

85m

1702

1m

QSR

QSR

QTSQSW m2

1m

For example,

if mQTS = 190, then

95m

1902

1m

QSW

QSW

S

R 1

W

X 2


DEPED COPY

188

O

N

L

M

K

P Q

S

R

Learn more about Tangents and Secants of a circle through the WEB. You may open the following links.

http://www.regentsprep.org/Regents/math/geometry/GP15/CircleAngles.htm http://www.math-worksheet.org/secant-tangent-angles http://www.mathopenref.com/tangentline.html http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/section/8.7/ http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/section/8.8/

Your goal in this section is to apply the key concepts of tangents

and secants of a circle. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided.

In the figure below, KL, KN, MP, and ML intersect Q at some points. Use the figure to answer the following questions.

5. Name all the intercepted arcs in the figure. Which angles intercept each of these arcs?

6. Suppose 50m KOM and m 130,KQM what is KLMm equal to? How about mNP?

Activity 4:

1. Which lines are tangent to the circle? Why?

2. Which lines are secants? Why?

3. At what points does each secant intersect the circle? How about the tangents?

4. Which angles are formed by two secant lines? two tangents? a tangent and a secant?


http://www.regentsprep.org/Regents/math/geometry/GP15/CircleAngles.htm


http://www.mathopenref.com/tangentline.html



DEPED COPY

189

Were you able to identify the tangents and secants in the figure, including the angles that they form? Were you able to identify the arcs that these angles intercept? Were you able to determine the unknown measure of the angle? I am sure you were! In the next activity, you will further apply the different ideas learned about tangents and secants in finding the measures of angles, arcs, and segments in some geometric figures.

Use the figure and the given information to answer the questions that follow. Explain how you arrived at your answer.

1. If mADC = 160 and mEF = 80, 2. If mMKL = 220 and mML = 140, what is m ?ABC what is m ?MQL

3. If mPR = 45 and mQS = 49, 4. Suppose mCG = 6x + 5,

what is m ?PTR m ?RTS mAR = 4x + 15, and 120m AEC . Find: a) x b) mCG c) mAR

Activity 5:

A

B C

D E

F

G

R C

A

E

P

S

R

Q

T


DEPED COPY

190

K O C

R

S

N

Q P

O

R

A S

P

B

Q

5. If mLGC = 149 and 39m LSC , 6. OK is tangent to R at C.

What is mMC? Suppose OCKC , OK = 56, and RC = 24. Find: OR, RS,

and KS.

7. If mQNO = 238, what is 8. PR is a diameter of O and

m ?PQO m ?PQR mRW = 55. Find:

9. Circles P and Q are tangent to each other at point S.

AB is tangent to both P and Q at S. Suppose AB = 16, AP = 12, and AQ = 10. What is the length

of PQ if it bisects AB ?

a. mPW d. WREm b. RPWm e. WERm c. PRWm f. EWRm

P R

E W

O

C

S

M

G

L


DEPED COPY

191

J R

A

T

K

10. AT is tangent to both circles K and J at A. ST

is tangent to K at S and RT is tangent to J at R. If 72 xST and

13 xRT , find: a. x c. RT

b. ST d. AT

How was the activity you have just done? Was it easy for you to determine the measures of the different angles, arcs, and segments? It was easy for sure!

In this section, the discussion was about the different geometric relationships involving tangents and secants of a circle.


deeper and move on to the next section.

Your goal in this section is to think deeper and test further your

understanding of the different geometric relationships involving tangents and secants of a circle. After doing the following activities, you should be able to find out how the different geometric relationships involving tangents and secants of a circle facilitate finding solutions to real-life problems and making wise decisions.

S


DEPED COPY

192

Answer the following.

1. In the figure on the right, RO and DN are tangent to U at O and N respectively. a. What is the measure of ?RON ?DNO

Explain how you arrived at your answer. b. Suppose NDUONR . Which angle is

congruent to ?NRO Why? c. If 31m ONR , what is NROm ? d. If m 49,DUN what is NDUm ? How about DUOm ? e. Suppose OU = 6, RN = 13, and ONDN , what is RO equal to? How

about DN? DU?

Is DUNNRO ? Justify your answer. 2. In the figure on the right, is LU tangent

to I? Why? How about SC? Justify your answer.

3. LR and LI are tangents to T from an external point L. a. Is RL congruent to ?LI Why?

b. Is ∆LTR congruent to ∆LTI? Justify

your answer.

c. Suppose m 38.RLT What is ILTm equal to?

How about m ?ITL m ?RTL

d. If 10RT and 24RL , what is the length of ?TL

How about the length of ?LI ?AL

Activity 6:

A

I

T

R

L

L

U 3

53

C

4

I

S

A

8

6

R O

N

U

D


DEPED COPY

193

C

D Z

S

M

Y X

4. In the figure on the right, ∆CDS is circumscribed about M. Suppose the perimeter of ∆CDS is 33 units, SX = 6 units, and DY = 3 units. What are the lengths of the following segments? Explain how you arrived at your answer.

a. SZ c. CX

b. DZ d. CY

5. From the main entrance of a park, there are two pathways where visitors

can walk along going to the circular garden. The pathways are both tangent to the garden whose center is 40 m away from the main entrance. If the area of the garden is about 706.5 m2, how long is each pathway?

6. The map below shows that the waters within ARC, a 250° arc, is

dangerous for shipping vessels. In the diagram, two lighthouses are located at points A and C and points P, R, and S are the locations of the ship at a certain time, respectively. a. What are the possible measures of ,P

,R and S ?

b. If you were the captain of a ship, how

would you make sure that your ship is in safe water?

Main Entrance

Garden

A

R C

P shore

S


DEPED COPY

194

How was the activity you have just performed? Did you gain better understanding of the lesson? Were you able to use the mathematics concepts and principles learned in solving problems? Were you able to realize the importance of the lesson in the real world? I am sure you were! In the next activity you will be proving geometric relationships involving tangents and secants.

Show a proof of the following theorems involving tangents and secants. 1. If a line is tangent to a circle, then it is perpendicular to the radius drawn to

the point of tangency. Given: AB is tangent to C at D. Prove: CDAB

2. If a line is perpendicular to a radius of a circle at its endpoint that is on the

circle, then the line is tangent to the circle. Given: RS is a radius of S. RSPQ Prove: PQ is tangent to S at R.

Activity 7:

R

Q

S P

D

C B

A


DEPED COPY

195

3. If two segments from the same exterior point are tangent to a circle, then the two segments are congruent.

Given: EM and EL are tangent to S at M and L, respectively.

Prove: EM EL

4. If two tangents, a secant and a tangent, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. a. Given: RS and TS are tangent to V

at R and T, respectively, and intersect at the exterior point S.

Prove: mTRmTQRRSTm 2

1

b. Given: KL is tangent to O at K. NL is a secant that passes through O at M and N. KL and NL intersect at the exterior point L.

Prove: mMKmNPKKLNm 2

1

c. Given: AC is a secant that passes through T at A and B. EC is a secant that passes through T at E and D. AC and EC intersect at the exterior point C.

Prove: mBDmAEACEm 2

1

L

E

S

M

Q

T

V S

R

M

K

L

N

P O

E

B

C T

A

D


DEPED COPY

196

5. If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Given: AC and EC are secants intersecting in the interior of V at T. PS and QR are the intercepted

arcs of PTS and .QTR

Prove: mQRmPSPTSm 2

1

6. If a secant and a tangent intersect at the point of tangency, then the

measure of each angle formed is one-half the measure of its intercepted arc. Given: MP and LN are secant and tangent, respectively, and intersect at C at the point of tangency, M.

Prove: mMPNMPm2

1 and

mMKPLMPm2

1

Were you able to prove the different geometric relationships involving tangents and secants? Were you convinced that these geometric relationships are true? I know you were! Find out by yourself how these geometric relationships are illustrated or applied in the real world.

In this section, the discussion was about your understanding of the different geometric relationships involving tangents and secants and how they are illustrated in real life.

What new realizations do you have about the different geometric relationships involving tangents and secants? How would you connect this to real life? How would you use this in making wise decisions?

M

K

L

N

P O

S

P

R

V

Q T


DEPED COPY

197



situations. You will be given a practical task which will demonstrate your understanding of the different geometric relationships involving tangents and secants.

Answer the following. Use the rubric provided to rate your work. 1. The chain and gears of bicycles or motorcycles or belt around two pulleys are

some real-life illustrations of tangents and circles. Using these real-life objects or similar ones, formulate problems involving tangents, then solve.

2. The picture below shows a bridge in the form of an arc. It also shows how secant is illustrated in real life. Using the bridge in the picture and other real-life objects, formulate problems involving secants, then solve them.

Activity 8:


DEPED COPY

198


Score Descriptors

6

Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes and provides explanations wherever appropriate.

5

Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.


3




Source: D.O. #73, s. 2012

In this section, your task was to formulate then solve problems involving the different geometric relationships involving tangents and secants.


SUMMARY/SYNTHESIS/GENERALIZATION:

This lesson was about different geometric relationships involving tangents and secants and their applications in real life. The lesson provided you with opportunities to find the measures of angles formed by secants and tangents and the arcs that these angles intercept. You also applied these relationships involving tangents and secants in finding the lengths of segments in some geometric figures. You were also given the opportunities to formulate and solve real-life problems involving tangents and secants of a circle. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning in the succeeding lessons.


DEPED COPY

199

Start Lesson 2B of this module by assessing your knowledge of the

different mathematical concepts previously studied and mathematical skills learned. These knowledge and skills will help you understand the relationships among tangent and secant segments. As you go through this lesson, think of this important question: “How do the relationships among tangent and secant segments facilitate finding solutions to real-life problems and making decisions?” To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier. You may check your work with your teacher.

Solve the following equations. Answer the questions that follow.

1. 3 27x 6. 2 25x 2. 4 20x 7. 2 64x 3. 1236 x 8. 2 12x 4. x763 9. 2 45x 5. x10158 10. 2 80x

a. How did you find the value of x in each equation? b. What mathematics concepts or principles did you apply in

solving the equations?

Were you able to find the value of x in each equation? Were you able to recall how the equations are solved? The skill applied in the previous activity will be used as you go on with the module.

Activity 1:


DEPED COPY

200

N S

E T

J

L

A

Use the figure below to answer the following questions.

1. Which of the lines or line segments is a tangent? secant? chord? Name these lines or line segments.

2. AT intersects LN at E. What are the

different segments formed? Name these segments.

3. What other segments can be seen in the figure? Name these segments.

4. SJ and LJ intersect at point J. How would you describe point J in relation to the given circle?

Was it easy for you to identify the tangent and secant lines and

chords and to name all the segments? I am sure it was! This time, find out the relationships among tangent, and secant segments, and external secant segments of circles by doing the next activity.

Perform the following activity.

Procedure: 1. In the given circle below, draw two intersecting chords BT and MN.

Activity 3:

Activity 2:


DEPED COPY

201

2. Mark and label the point of intersection of the two chords as A. 3. With a ruler, measure the lengths of the segments formed by the

intersecting chords.

What is the length of each of the following segments? a. BA c. MA

b. TA d. NA

4. Compare the product of BA and TA with the product of MA and NA. 5. Repeat #1 to #4 using other pairs of chords of different lengths.

What conclusion can you make?

Were you able to determine the relationship that exists among

segments formed by intersecting chords of a circle? For sure you were able to do it. In the next activity, you will see how tangent and secant segments are used in real-life situations.

Use the situation below to answer the questions that follow.

You are in a hot air balloon and your eye level is 60 meters over the

ocean. Suppose your line of sight is tangent to the radius of the earth like the illustration shown below.

1. How far away is the farthest point you can see over the ocean if the radius

of the earth is approximately 6378 kilometers?

Activity 4:


DEPED COPY

202

E

A

M

G

S

In the figure, GM and SM are secants.

AM and EM are external secant segments.

2. What mathematics concepts would you apply to find the distance from where you are to any point on the horizon?

How did you find the preceding activities? Are you ready to learn about tangent and secant segments? I am sure you are! From the activities done, you were able to find out how tangent and secant segments of circles are illustrated in real life. But how do the relationships among tangent and secant segments of circles facilitate finding solutions to real-life problems and making decisions? You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on tangent and secant segments of circles and the examples presented.

Theorem on Two Intersecting Chords

If two chords of a circle intersect, then the product of the measures of

the segments of one chord is equal to the product of the measures of the segments of the other chord.

External Secant Segment

An external secant segment is the part of a secant segment that is

outside a circle.

In the circle shown on the right, SN

intersects DL at A. From the theorem, LADANASA .

S

D

L

A

N


DEPED COPY

203

N

A R

I

E

N

O C

Y

Theorems on Secant Segments, Tangent Segments, and External Secant Segments

1. If two secant segments are drawn to a circle from an exterior point, then

the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.

2. If a tangent segment and a secant segment are drawn to a circle from an

exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.

Learn more about Tangent and Secant Segments of a Circle through the WEB. You may open the following links.


http://www.cliffsnotes.com/math/geometry/circles/segments-of-chords-secants-tangents

http://www.mathopenref.com/secantsintersecting.html


http://www.math-worksheet.org/tangents

AR and NR are secant segments drawn to the circle from an exterior point R. From the theorem, .AR IR NR ER

YO is a secant segment drawn to the circle from exterior point O. CO is a tangent segment that is also drawn to the circle from the same exterior point O. From the theorem,

2 .CO YO NO








DEPED COPY

204

Your goal in this section is to apply the key concepts of tangent and

secant segments of a circle. Use the mathematical ideas and the examples presented in the preceding section to answer the given activities.

Name the external secant segments in each of the following figures.

1. 4.

2. 5.

3. 6.

Were you able to identify the external secant segments in the given circles? In the next activity, you will apply the theorems you have learned in this lesson.

Activity 5:

E

L

R

O

W F

I L Y

M

E

T

D S

M

C

E

O

S

J

G

R

I

L

I F

A D

B

J E

H G

K C


DEPED COPY

205

Find the length of the unknown segment (x) in each of the following figures. Answer the questions that follow.

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

Activity 6:

6 5

4 3

x

L

G

A F S

4 6 5

12 x D

I

U

G

E

8 7 5

12 x

O M

A R

N

5 6 5

4 x T

E

U J

N

T

I H S

F 10

8 5

x

16

9 16

x

A

S

O R

S

N

A E

J

4

11 5 12

x x

C

T

M

S

6

4 5

6

8 5

6

x G

C

A M

I

10

O

E

L

x

25 5

V


DEPED COPY

206

Questions: a. How did you find the length of the unknown segment?

What geometric relationships or theorems did you apply to come up with your answer?

b. Compare your answers with those of your classmates. Did you arrive at the same answer? Explain.

In the activity you have just done, were you able to apply the theorems

you have learned? I am sure you were! In the next activity, you will use the theorems you have studied in this lesson.

Answer the following. 1. Draw and label a circle that fits the following descriptions.

a. has center L b. has secant segments MO and QO c. has external secant segments NO and PO d. has tangent segment RO

2.

How was the activity you have just done? Was it easy for you to apply the theorems on secant segments and tangent segments? It was easy for sure!

In this section, the discussion was about tangent and secant segments and their applications in solving real-life problems.



deeper by moving on to the next section.

Activity 7:

In the figure on the right, SU and WU are secant segments and XU is a tangent segment. If 14WU ,

12ST , and 4TU , find:

a. VU

b. XU

S

T W

X

V

14 5

4 5

12 5

U


DEPED COPY

207

Your goal in this section is to take a closer look at some aspects of the topic. You are going to think deeper and test further your understanding of tangents and secant segments. After doing the following activities, you should be able to answer this important question: How do tangents and secant segments of circles facilitate finding solutions to real-life problems and making decisions?

Show a proof of each of the following theorems. 1. If two chords of a circle intersect, then the product of the measures of the

segments of one chord is equal to the product of the measures of the segments of the other chord.

2. If two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.

Activity 8:

Given: AB and DE are chords of C intersecting at M. Prove: EMDMBMAM

A

D

B

E C

M

Given: DP and DS are secant segments of T drawn from exterior point D. Prove: DRDSDQDP

P

T

S D

Q

R


DEPED COPY

208

3. If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.

Were you able to prove the theorems on intersecting chords, secant

segments, and tangent segments? I am sure you did!

Let us find out more about these theorems and their applications. Perform the next activity.

Answer the following questions. 1. Jurene and Janel were asked to find the length of AB in the figure below.

The following are their solutions.

Who do you think would arrive at the correct answer? Explain your answer.

Activity 9:

Jurene: 7 9 10x

Janel: 7 7 9 9 10x 7

x

9 10

B

D C

A

E

Given: KL and KM are tangent and secant segments, respectively, of O drawn

from exterior point K. KM intersects O at N. Prove: KNKMKL

2

K

L

O

M N


DEPED COPY

209

2. The figure below shows a sketch of a circular children’s park and the different pathways from the main road. If the distance from the main road to Gate 2 is 70 m and the length of the pathway from Gate 2 to the Exit is 50 m, about how far from the main road is Gate 1?

3.

b. Suppose Anton hangs 40 pairs of light balls on the ceiling of a hall in preparation for an event. How long is the string that he needs to hang these light balls if each has a diameter of 12 cm and the point of tangency of each pair of balls is 30 cm from the ceiling?

How did you find the activity? Were you able to find out some real-

life applications of the different geometric relationships involving tangents and secant segments? Do you think you could cite some more real-life applications of these? I am sure you could. Try doing the next activity.

Anton used strings to hang two small light balls on the ceiling as shown in the figure on the right. The broken line represents the distance from the point of tangency of the two light balls to the ceiling.

a. Suppose the diameter of each light ball is 10 cm and the length of the string used to hang it is 40 cm. How far is the point of tangency of the two light balls from the ceiling?

Exit

Gate 1

Gate 2

Gate 3

Main Road


DEPED COPY

210

In this section, the discussion was about your understanding of tangent and secant segments and how they are used in real life.

What new realizations do you have about tangent and secant segments? How would you connect this to real life?

Now that you have a deeper understanding of the topic, you are

ready to do the tasks in the next section.

Your goal in this section is to apply your learning to real-life situations. You will be given a practical task which will demonstrate your understanding of tangent and secant segments.

Make a design of an arch bridge that would connect two places which are separated by a river, 20 m wide. Indicate on the design the different measurements of the parts of the bridge. Out of the design and the measurements of its parts, formulate problems involving tangent and secant segments, and then solve. Use the rubric provided to rate your work. Rubric for the Bridge’s Design

Score Descriptors

4 The bridge’s design is accurately made, presentable, and appropriate.

3 The bridge’s design is accurately made and appropriate but not presentable.

2 The bridge’s design is not accurately made but appropriate. 1 The bridge’s design is made but not appropriate.

Activity 10:


DEPED COPY

211


Score Descriptors

6


5

Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

4

Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

3


2

Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension.


Source: D.O. #73, s. 2012

In this section, your task was to formulate problems where tangent and secant segments of circles are illustrated.



This lesson was about the geometric relationships involving tangent

and secant segments. In this lesson, you were able to find the lengths of segments formed by tangents and secants. You were also given the opportunity to design something practical where tangent and secant segments are illustrated or applied. Then, you were asked to formulate and solve problems out of this design. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the succeeding lessons in mathematics.


DEPED COPY

212

GLOSSARY OF TERMS Arc – a part of a circle

Arc Length – the length of an arc which can be determined by using the

proportion r

lA

2360, where A is the degree measure of this arc, r is the

radius of the circle, and l is the arc length

Central Angle – an angle formed by two rays whose vertex is the center of the circle

Common External Tangents – tangents which do not intersect the segment joining the centers of the two circles

Common Internal Tangents – tangents that intersect the segment joining the centers of the two circles

Common Tangent – a line that is tangent to two circles on the same plane

Congruent Arcs – arcs of the same circle or of congruent circles with equal measures

Congruent Circles – circles with congruent radii

Degree Measure of a Major Arc – the measure of a major arc that is equal to 360 minus the measure of the minor arc with the same endpoints

Degree Measure of a Minor Arc – the measure of the central angle which intercepts the arc

External Secant Segment – the part of a secant segment that is outside a circle Inscribed Angle – an angle whose vertex is on a circle and whose sides contain chords of the circle

Intercepted Arc – an arc that lies in the interior of an inscribed angle and has endpoints on the angle

Major Arc – an arc of a circle whose measure is greater than that of a semicircle


DEPED COPY

213

Minor Arc – an arc of a circle whose measure is less than that of a semicircle

Point of Tangency – the point of intersection of the tangent line and the circle

Secant – a line that intersects a circle at exactly two points. A secant contains a chord of a circle

Sector of a Circle – the region bounded by an arc of the circle and the two radii to the endpoints of the arc

Segment of a Circle – the region bounded by an arc and the segment joining its endpoints

Semicircle – an arc measuring one-half the circumference of a circle

Tangent to a Circle – a line coplanar with the circle and intersects it at one and only one point

LIST OF THEOREMS AND POSTULATES ON CIRCLES Postulates: 1. Arc Addition Postulate. The measure of an arc formed by two adjacent

arcs is the sum of the measures of the two arcs.

2. At a given point on a circle, one and only one line can be drawn that is tangent to the circle.

Theorems: 1. In a circle or in congruent circles, two minor arcs are congruent if and only

if their corresponding central angles are congruent.

2. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

3. In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is perpendicular to the chord.

4. If an angle is inscribed in a circle, then the measure of the angle equals

one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle).


DEPED COPY

214



right angle.

7. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

8. If a line is tangent to a circle, then it is perpendicular to the radius drawn to

the point of tangency.

9. If a line is perpendicular to a radius of a circle at its endpoint that is on the circle, then the line is tangent to the circle.

10. If two segments from the same exterior point are tangent to a circle, then

the two segments are congruent.

11. If two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

12. If a secant and a tangent intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

13. If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

14. If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

15. If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc.

16. If two chords of a circle intersect, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord.


DEPED COPY

215

17. If two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.

18. If a tangent segment and a secant segment are drawn to a circle from an

exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.

DepEd Instructional Materials That Can Be Used as Additional Resources 1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third

Year Mathematics. Module 18: Circles and Their Properties.

2. Distance Learning Module (DLM) 3, Modules 1 and 2: Circles.

REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: References: Bass, Laurie E., Randall, I. Charles, Basia Hall, Art Johnson, and Kennedy,

D. Texas Geometry. Pearson Prentice Hall, Boston, Massachusetts 02116, 2008.

Bass, Laurie E., Rinesmith Hall B., Johnson A., and Wood, D. F. Prentice Hall

Geometry Tools for a Changing World. Prentice-Hall, Inc., NJ, USA, 1998.

Boyd, Cummins, Malloy, Carter, and Flores. Glencoe McGraw-Hill Geometry.

The McGraw-Hill Companies, Inc., USA, 2008. Callanta, Melvin M. Infinity, Worktext in Mathematics III. EUREKA Scholastic

Publishing, Inc., Makati City, 2012. Chapin, Illingworth, Landau, Masingila, and McCracken. Prentice Hall Middle

Grades Math, Tools for Success, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1997.

Cifarelli, Victor, et al. cK-12 Geometry, Flexbook Next Generation Textbooks,

Creative Commons Attribution-Share Alike, USA, 2009.


DEPED COPY

216

Clemens, Stanley R., Phares G. O’Daffer, Thomas J. Cooney, and John A.

Dossey. Addison-Wesley Geometry. Addison-Wesley Publishing Company, Inc., USA, 1990.

Clements, D. H., Jones, K. W., Moseley, L. B., and Schulman, L. Math in My

World, McGraw-Hill Division, Farmington, New York, 1999. Department of Education. K to 12 Curriculum Guide Mathematics,

Department of Education, Philippines, 2012. Gantert, Ann Xavier. AMSCO’s Geometry. AMSCO School Publications, Inc.,

NY, USA, 2008. Renfro, Freddie L. Addison-Wesley Geometry Teacher’s Edition. Addison-

Wesley Publishing Company, Inc., USA, 1992. Rich, Barnett and Christopher Thomas. Schaum’s Outlines Geometry Fourth

Edition. The McGraw-Hill Companies, Inc., USA, 2009. Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and

Marvin L. Bittinger. Addison-Wesley Informal Geometry. Addison-Wesley Publishing Company, Inc., USA, 1992.

Wilson, Patricia S., et al. Mathematics, Applications and Connections, Course

I, Glencoe Division of Macmillan/McGraw-Hill Publishing Company, Westerville, Ohio, 1993.


DEPED COPY

217

Website Links as References and Sources of Learning Activities: CK-12 Foundation. cK-12 Inscribed Angles. (2014). Retrieved June 29, 2014, from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.7/ CK-12 Foundation. cK-12 Secant Lines to Circles. (2014). Retrieved June 29, 2014, from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.8/ CK-12 Foundation. cK-12 Tangent Lines to Circles. (2014). Retrieved June 29, 2014, from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.4/ Houghton Mifflin Harcourt. CliffsNotes. Arcs and Inscribed Angles. (2013). Retrieved June 29, 2014, from http://www.cliffsnotes.com/math/geometry/ circles/arcs-and-inscribed-angles Houghton Mifflin Harcourt. CliffsNotes. Segments of Chords, Secants, and Tangents. (2013). Retrieved June 29, 2014, from http://www.cliffsnotes.com/math/geometry/circles/segments-of-chords-secants-tangents Math Open Reference. Arc. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/arc.html Math Open Reference. Arc Length. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/arclength.html Math Open Reference. Central Angle. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/circlecentral.html Math Open Reference. Central Angle Theorem. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/arccentralangletheorem.html Math Open Reference. Chord. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/chord.html Math Open Reference. Inscribed Angle. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/circleinscribed.html

Math Open Reference. Intersecting Secants Theorem. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/secantsintersecting.html


http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/%20section/8.7/






http://www.cliffsnotes.com/

http://www.cliffsnotes.com/

http://www.mathopenref.com/arclength.html

http://www.mathopenref.com/arccentralangletheorem.html



DEPED COPY

218

Math Open Reference. Sector. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/arcsector.html Math Open Reference. Segment. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/segment.html math-worksheet.org. Free Math Worksheets. Arc Length and Sector Area. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/arc-length-and-sector-area math-worksheet.org. Free Math Worksheets. Inscribed Angles. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/inscribed-angles math-worksheet.org. Free Math Worksheets. Secant-Tangent Angles. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/secant-tangent-angles math-worksheet.org. Free Math Worksheets. Tangents. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/tangents OnlineMathLearning.com. Circle Theorems. (2013). Retrieved June 29, 2014, from http://www.onlinemathlearning.com/circle-theorems.html Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Formulas for Angles in Circles Formed by Radii, Chords, Tangents, Secants. (2012). Retrieved June 29, 2014, from http://www.regentsprep.org/Regents/math/geometry/ GP15/CircleAngles.htm

Website Links for Videos: Coach, Learn. NCEA Maths Level 1 Geometric reasoning: Angles Within Circles. (2012). Retrieved June 29, 2014, from http://www.youtube.com/watch?v=jUAHw-JIobc Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved June 29, 2014, from https://www.khanacademy.org/math/geometry/cc-geometry-circles Schmidt, Larry. Angles and Arcs Formed by Tangents, Secants, and Chords. (2013).Retrieved June 29, 2014, from http://www.youtube.com/watch?v=I-RyXI7h1bM Sophia.org. Geometry. Circles. (2014). Retrieved June 29, 2014, from http://www.sophia.org/topics/circles


http://www.mathopenref.com/arcsector.html

http://www.mathopenref.com/segment.html





http://www.math-worksheet.org/secant-tangent-angles

http://www.math-worksheet.org/secant-tangent-angles

http://www.onlinemathlearning.com/circle-theorems.html

http://www.regentsprep.org/Regents/math/geometry/%20GP15/CircleAngles.htm

DEPED COPY

219

Website Links for Images: Cherry Valley Nursery and Landscape Supply. Seasonal Colors Flowers and

Plants. (2014). Retrieved June 29, 2014 from http://www.cherryvalleynursery.com/

eBay Inc. Commodore Holden CSA Mullins pursuit mag wheel 17 inch

genuine - 4blok #34. (2014). Retrieved June 29, 2014, from

http://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag-

wheel-17-inch-genuine-4blok-34-/221275049465

Fort Worth Weekly. Facebook Fact: Cowboys Are World’s Team. (2012) . Retrieved June 29, 2014 from http://www.fwweekly.com/2012/08/21/

facebook-fact-cowboys-now-worlds-team/

GlobalMotion Media Inc. Circular Quay, Sydney Harbour to Historic Hunter's

Hill Photos. (2013). Retrieved June 29, 2014 from http://www.everytrail.com/

guide/circular-quay-sydney-harbour-to-historic-hunters-hill/photos

HiSupplier.com Online Inc. Shandong Sun Paper Industry Joint Stock Co.,Ltd.

Retrieved June 29, 2014, from http://pappapers.en.hisupplier.com/product-

66751-Art-Boards.html

Kable. Slip-Sliding Away. (2014). Retrieved June 29, 2014, from

http://www.offshore-technology.com/features/feature1674/feature1674-5.html

Materia Geek. Nikon D500 presentada officialmente. (2009). Retrieved June

29, 2014 from http://materiageek.com/2009/04/nikon-d5000-presentada-

oficialmente/

Piatt, Andy. Dreamstime.com. Rainbow Stripe Hot Air Balloon. Retrieved

June 29, 2014, from http://thumbs.dreamstime.com/z/rainbow-stripe-hot-air-

balloon-788611.jpg

Regents of the University of Colorado. Nautical Navigation. (2014). Retrieved

June 29, 2014, from http://www.teachengineering.org/view_activity.php?url=

collection/cub_/activities/cub_navigation/cub_navigation_lesson07_activity1.xml


http://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag-wheel-17-inch-genuine-4blok-34-/221275049465

http://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag-wheel-17-inch-genuine-4blok-34-/221275049465

http://www.globalmotion.com/

http://www.everytrail.com/

http://www.offshore-technology.com/features/feature1674/feature1674-5.html

http://materiageek.com/2009/04/nikon-d5000-presentada-oficialmente/

http://materiageek.com/2009/04/nikon-d5000-presentada-oficialmente/

http://thumbs.dreamstime.com/z/rainbow-stripe-hot-air-balloon-788611.jpg

http://thumbs.dreamstime.com/z/rainbow-stripe-hot-air-balloon-788611.jpg

http://www.teachengineering.org/view_activity.php?url

DEPED COPY

220

Sambhav Transmission. Industrial Pulleys. Retrieved June 29, 2014 from

http://www.indiamart.com/sambhav-transmission/industrial-pulleys.html

shadefxcanopies.com. Flower Picture Gallery, Garden Pergola Canopies. Retrieved June 29, 2014, from http://www.flowerpicturegallery.com/v/halifax-public-gardens/Circular+mini+garden+with+white+red+flowers+and+ dark+grass+in+the+middle+at+Halifax+Public+Gardens.jpg.html

Tidwell, Jen. Home Sweet House. (2012). Retrieved June 29, 2014 from

http://youveneverheardofjentidwell.com/2012/03/02/home-sweet-house/

Weston Digital Services. FWR Motorcycles LTD. CHAINS AND

SPROCKETS. (2014). Retrieved June 29, 2014 from

http://fwrm.co.uk/index.php?main_page=index&cPath=585&zenid=10omr4he

hmnbkktbl94th0mlp6


http://youveneverheardofjentidwell.com/2012/03/02/home-sweet-house/

DEPED COPY

221

I. INTRODUCTION

Look around! What geometric figures do you see in your classroom, school buildings, houses, bridges, roads, and other structures? Have you ever asked yourself how geometric figures helped in planning the construction of these structures?

In your community or province, was there any instance when a

stranger or a tourist asked you about the location of a place or a landmark? Were you able to give the right direction and how far it is? If not, could you give the right information the next time somebody asks you the same question?

Find out the answers to these questions and determine the vast applications of plane coordinate geometry through this module.


DEPED COPY

222

II. LESSONS AND COVERAGE:

In this module, you will examine the questions asked in the preceding page when you take the following lessons:

Lesson 1 – The Distance Formula, The Midpoint, and The Coordinate Proof

Lesson 2 – The Equation of a Circle In these lessons, you will learn to:

Lesson 1

derive the distance formula; apply the distance formula in proving some geometric

properties; graph geometric figures on the coordinate plane; and solve problems involving the distance formula.

Lesson 2

illustrate the center-radius form of the equation of a circle; determine the center and radius of a circle given its

equation and vice versa; graph a circle on the coordinate plane; and solve problems involving circles on the coordinate plane.

Here is a simple map of the lessons that will be covered in this module:

Plane Coordinate Geometry

Problems Involving Geometric Figures on the Coordinate

Plane

The Distance Formula The Midpoint Formula

Coordinate Proof

The Equation and Graph of a Circle


DEPED COPY

223

III. PRE-ASSESSMENT

Part I

Find out how much you already know about this module. Choose the letter that you think best answers each of the following questions. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module.

1. Which of the following represents the distance d between the two

points 1 1,x y and 2 2

,x y ?

A. 2 2

2 1 2 1d x x y y C.

2 2

2 1 2 1d x x y y

B. 2 2

2 1 2 1d x x y y D.

2 2

2 1 2 1d x x y y

2. Point L is the midpoint of KM . Which of the following is true about the distances among K, L, and M? A. KMKL C. LMKL

B. KMLM D. LMKLKM 2

3. A map is drawn on a grid where 1 unit is equivalent to 1 km. On the

same map, the coordinates of the point corresponding to San Vicente is (4, 9). Suppose San Vicente is 13 km away from San Luis. Which of the following could be the coordinates of the point corresponding to San Luis? A. (-13, 0) B. (16, 4) C. (4, 16) D. (0, 13)

4. What is the distance between the points M(-3,1) and N(7,-3)?

A. 6 B. C. 14 D.

5. Which of the following represents the midpoint M of the segment

whose endpoints are 1 1,x y and 2 2

,x y ?

A. 1 2 1 2,2 2

x x y yM

C. 1 1 2 2,2 2

x y x yM

B. 1 2 1 2,2 2

x x y yM

D. 1 1 2 2,2 2

x y x yM

6. What are the coordinates of the midpoint of a segment whose

endpoints are (-1, -3) and (11, 7)? A. (2, 5) B. (6, 5) C. (-5, -2) D. (5, 2)


DEPED COPY

224

7. Which of the following equations describe a circle on the coordinate plane with a radius of 4 units?

A. 2 2 24 4 2x y C.

2 2 22 2 4x y

B. 2 2 22 2 4x y D.

2 2 24 4 16x y

8. P and Q are points on the coordinate plane as shown in the figure below.

If the coordinates of P and Q are 52, and 58, , respectively, which

of the following would give the distance between the two points?

A. 52 B. 58 C. 28 D. 82

9. A new transmission tower will be put up midway between two existing towers. On a map drawn on a coordinate plane, the coordinates of the first existing tower are (–5, –3) and the coordinates of the second existing tower are (9,13). What are the coordinates of the point where the new tower will be placed? A. (2, 5) B. (7, 8) C. (4, 10) D. (14, 16)

10. What proof uses figures on a coordinate plane to prove geometric properties? A. indirect proof C. coordinate proof B. direct proof D. two-column proof

11. The coordinates of the vertices of a square are H(3, 8), I(15, 8), J(15, –4), and K(3, –4). What is the length of a diagonal of the square?

A. 4 B. 8 C. 12 D. 212

12. The coordinates of the vertices of a triangle are T(–1, –3), O(7, 5), and

P(7, –2). What is the length of the segment joining the midpoint of OT and P?

A. 5 B. 4 C. 3 D. 7

y

x


DEPED COPY

225

13. What figure is formed when the points A(3, 7), B(11, 10), C(11, 5), and D(3, 2) are connected consecutively? A. parallelogram C. square B. trapezoid D. rectangle

14. In the parallelogram below, what are the coordinates of Q? A. (a, b+c) B. (a+b,c) C. (a-b,c) D. (a,b-c)

15. Diana, Jolina, and Patricia live in three different places. The location of their houses are shown on a coordinate plane below.

About how far is Jolina’s house from Diana’s house? A. 10 units B. 10.58 units C. 11.4 units D. 12 units

16. What is the center of the circle 2 2 4 10 13 0x y x y ?

A. (2, 5) B. (–2, 5) C. (2, –5) D. (–2, –5)

S(0, 0) R(b, 0)

P(a, c) Q

Diana

Jolina

Patricia

x

y


DEPED COPY

226

17. Point F is 5 units from point D whose coordinates are (6, 2). If the x-coordinate of F is 10 and lies in the first quadrant, what is its y-coordinate? A. -3 B. -1 C. 5 D. 7

18. The endpoints of a diameter of a circle are L(–3, –2) and G(9, –6). What is the length of the radius of the circle?

A. 10 B. 2 10 C. 4 10 D. 108

19. A radius of a circle has endpoints (4, –1) and (8, 2). What is the equation that defines the circle if its center is at the fourth quadrant?

A. 2 2

8 2 25x y C. 2 2

8 2 100x y

B. 2 2

4 1 100x y D. 2 2

4 1 25x y

20. On a grid map of a province, the coordinates that correspond to the location of a cellular phone tower is (–2, 8) and it can transmit signals up to a 12 km radius. What is the equation that represents the transmission boundaries of the tower?

A. 2 2 4 16 76 0x y x y C. 2 2 4 16 76 0x y x y

B. 2 2 4 16 76 0x y x y D. 2 2 4 16 76 0x y x y

Part II

Solve each of the following problems. Show your complete solutions.

1. A tracking device in a car indicates that it is located at a point whose coordinates are (17, 14). In the tracking device, each unit on the grid is equivalent to 5 km. How far is the car from its starting point whose coordinates are (1, 2)?

2. A radio signal can transmit messages up to a distance of 3 km. If the radio signal’s origin is located at a point whose coordinates are (4, 9), what is the equation of the circle that defines the boundary up to which the messages can be transmitted?

Rubric for Problem Solving

Score Descriptors

4 Used an appropriate strategy to come up with a correct solution and arrived at a correct answer.

3 Used an appropriate strategy to come up with a solution. But a part of the solution led to an incorrect answer.

2 Used an appropriate strategy but came up with an entirely wrong solution that led to an incorrect answer.

1 Attempted to solve the problem but used an inappropriate strategy that led to a wrong solution.


DEPED COPY

227

Part III Read and understand the situation below, then answer the question or perform what are asked. The Scout Master of your school was informed that the Provincial Boy Scouts Jamboree will be held in your municipality. He was assigned to prepare the area that will accommodate the delegates from 30 municipalities. It is expected that around 200 boy scouts will join the jamboree from each municipality.

To prepare for the event, he made an ocular inspection of the area where the jamboree will be held. The area is rectangular in shape and is large enough for the delegates to set up their tents and other camping structures. Aside from these, there is also a provision for the jamboree headquarter, medics quarter in case of emergency and other health needs, walkways, and roads, security posts, and a large ground where the different boy scouts events will be held. Aside from conducting an ocular inspection, he was also tasked to prepare a large ground plan to be displayed in front of the camp site. Copies of the ground plan will also be given to heads of the different delegations.

1. Suppose you are the Scout Master, how will you prepare the ground plan of the Boy Scouts jamboree?

2. Prepare the ground plan. Use a piece of paper with a grid and coordinate axes. Indicate the scale used.

3. On the grid paper, indicate the proposed locations of the different delegations, the jamboree headquarter, medics quarter, walkways and roads, security posts, and the boy scouts event ground.

4. Determine all the mathematics concepts or principles already learned that are illustrated in the prepared ground plan.

5. Formulate equations and problems involving these mathematics concepts or principles, then solve.

Rubric for Ground Plan

Score Descriptors

4 The ground plan is accurately made, appropriate, and presentable.

3 The ground plan is accurately made and appropriate but not presentable.

2 The ground plan is not accurately made but appropriate.

1 The ground plan is not accurately made and not appropriate.


DEPED COPY

228

Rubric for Equations Formulated and Solved

Score Descriptors 4 All equations are properly formulated and solved correctly.

3 All equations are properly formulated but some are not solved correctly.

2 All equations are properly formulated but at least 3 are not solved correctly.

1 All equations are not properly formulated and solved.


Score Descriptors

6

Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate

5

Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes

4

Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes

3

Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details

2

Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension

1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach

Source: D.O. #73, s. 2012

IV. LEARNING GOALS AND TARGETS:

After going through this module, you should be able to demonstrate understanding of key concepts of plane coordinate geometry particularly the distance formula, equation of a circle, and the graphs of circles and other geometric figures. Also, you should be able to formulate and solve problems involving geometric figures on the rectangular coordinate plane.


DEPED COPY

229

Start Lesson 1 of this module by assessing your knowledge of the

different mathematical concepts previously studied and your skills in performing mathematical operations. These knowledge and skills will help you understand the distance formula. As you go through this lesson, think of this important question: How do the distance formula, the midpoint formula, and the coordinate proof facilitate finding solutions to real-life problems and making wise decisions? To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier. You may check your work with your teacher.

Use the number line below to find the length of each of the following segments and then answer the questions that follow.

1. AB 4. DE

2. BC 5. EF

3. CD 6. FG

Questions: 1. How did you find the length of each segment? 2. Did you use the coordinates of the points in finding the length of each

segment? If yes, how? 3. Which segments are congruent? Why? 4. How would you relate the lengths of the following segments?

d.1) AB , BC , and AC d.2) AC , CE , and AE

A G F E C BQ

D

Activity 1:


DEPED COPY

230

5. Is the length of AD the same as the length of DA? How about BF and FB ? Explain your answer.

Were you able to determine the length of each segment? Were you

able to come up with relationships among the segments based on their lengths? What do you think is the significance of this activity in relation to your new lesson? Find this out as you go through this module.

The length of one side of each right triangle below is unknown. Determine the length of this side. Explain how you obtained your answer. 1. 4.

2. 5.

3. 6.

In the activity, you have just done, were you able to determine the length of the unknown side of each right triangle? I know you were able to do it! The mathematics principles you applied in finding each unknown side is related to your new lesson, the distance formula. Do you know why? Find this out in the succeeding activities!

Activity 2:

6

? 4

8 12

?

? 24

18

3 ?

4

9 15

?

5 13

?


DEPED COPY

231

Use the situation below to answer the questions that follow. Jose lives 5 km away from the plaza. Every Saturday, he meets Emilio and Diego for a morning exercise. In going to the plaza, Emilio has to travel 6 km to the west while Diego has to travel 8 km to the south. The location of their houses and the plaza are illustrated on the coordinate plane as shown below.

1. How far is Emilio’s house from Diego’s house? Explain your answer. 2. Suppose the City Hall is 4 km north of Jose’s house. How far is it from

the plaza? from Emilio’s house? Explain your answer. 3. How far is the gasoline station from Jose’s house if it is km south of

Emilio’s house? Explain your answer. 4. What are the coordinates of the points corresponding to the houses of

Jose, Emilio, and Diego? How about the coordinates of the point corresponding to the plaza?

Activity 3:

Emilio’s house

Diego’s house

Plaza

y

x

City Hall

Gasoline Station

Jose’s house


DEPED COPY

232

5. If the City Hall is km north of Jose’s house, what are the coordinates of the point corresponding to it? How about the coordinates of the point corresponding to the gasoline station if it is km south of Emilio’s house?

6. How are you going to use the coordinates of the points in determining the distance between Emilio’s house and the City Hall? Jose’s house and the gasoline station? The distances of the houses of Jose, Emilio, and Diego from each other? Explain your answer.

Did you learn something new about finding the distance between two objects? How is it different from or similar with the methods you have learned before? Learn about the distance formula and its derivation by doing the next activity.

Perform the following activity. Answer every question that follows.

1. Plot the points A(2,1) and B(8,9) on the coordinate plane below. 2. Draw a horizontal line passing through A and a vertical line containing

B.

3. Mark and label the point of intersection of the two lines as C. What are the coordinates of C? Explain how you obtained your answer.

What is the distance between A and C?

How about the distance between B and C?

Activity 4:

x

y


DEPED COPY

233

4. Connect A and B by a line segment. What kind of triangle is formed by A, B, and C? Explain your answer. How will you find the distance between A and B? What is AB equal to?

5. Replace the coordinates of A by (x1, y1) and B by (x2, y2).

What would be the resulting coordinates of C? What expression represents the distance between A and C? How about the expression that represents the distance between B and C?

What equation will you use to find the distance between A and B? Explain your answer.

How did you find the preceding activities? Are you ready to learn about the distance formula and its real-life applications? I am sure you are! From the activities done, you were able to find the distance between two points or places using the methods previously learned. You were able to derive also the distance formula. But how does the distance formula facilitate solving real-life problems and making wise decisions? You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on the distance formula including the midpoint formula and the coordinate proof. Understand very well the examples presented so that you will be guided in doing the succeeding activities.


DEPED COPY

234

Distance between Two Points

The distance between two points is always nonnegative. It is positive when the two points are different, and zero if the points are the same. If P and Q are two points, then the distance from P to Q is the same as the distance from Q to P. That is, PQ = QP.

Consider two points that are aligned horizontally or vertically on the coordinate plane. The horizontal distance between these points is the absolute value of the difference of their x-coordinates. Likewise, the vertical distance between these points is the absolute value of the difference of their y-coordinates.

Example 1: Find the distance between P(3, 2) and Q(10, 2).

Solution:

Since P and Q are aligned horizontally, then 310PQ

or 7PQ .

Example 2: Determine the distance between A(4, 3) and B(4, –5).

Solution:

y

x

Q P

y

x

Points A and B are on the same vertical line. So the

distance between them is 53 AB . This can be

simplified to 53 AB or 8AB .


DEPED COPY

235

The Distance Formula The distance between two points, whether or not they are aligned

horizontally or vertically, can be determined using the distance formula. Consider the points P and Q whose coordinates are (x1, y1) and

(x2, y2), respectively. The distance d between these points can be determined

using the distance formula 2 2

2 1 2 1d x x y y or

2 2

2 1 2 1PQ x x y y .

Example 1: Find the distance between P(1, 3) and Q(7, 11). Solution: To find the distance between P and Q, the following

procedures can be followed.

1. Let 1 1,x y = (1, 3) and 2 2,x y = (7, 11).

2. Substitute the corresponding values of

1 1 2 2, , , andx y x y in the distance formula

2 2

2 1 2 1PQ x x y y .

y

x

P(x1, y1)

Q(x2, y2)

PQ


DEPED COPY

236

3. Solve the resulting equation.

2231117 PQ

2286

6436

100

10PQ

The distance between P and Q is 10 units.

Example 2: Determine the distance between A(1, 6) and B(5, –2).

Solution: Let 11 x , 61 y , 52 x , and 2y 2. Then substitute

these values in the formula 212

2

12 yyxxAB .

226215 AB

Simplify.

226215 AB

2284

6416

80

516

54AB or 948.AB

The distance between A and B is 54 units or

approximately 8.94 units. The distance formula has many applications in real life. In particular, it

can be used to find the distance between two objects or places. Example 3: A map showing the locations of different municipalities

and cities is drawn on a coordinate plane. Each unit on the coordinate plane is equivalent to 6 kilometers. Suppose the coordinates of Mabini City is (2, 2) and Sta. Lucia town is (6, 8). What is the shortest distance between these two places?


DEPED COPY

237

Solution: Let 1

2x , 1

2y , 2

6x , and 2

8.y Then substitute these

values into the distance formula 2 2

2 1 2 1d x x y y .

2 2

6 2 8 2d

Simplify the expression.

2 2

6 2 8 2d

2 2

4 6

16 36

52

2 13d units or 7.21d units

Since 1 unit on the coordinate plane is equivalent to 6 units, multiply the obtained value of d by 6 to get the distance between Sta. Lucia town and Mabini City.

7.21 6 43.26

The distance between Sta. Lucia town and Mabini City is approximately 43.26 km.


DEPED COPY

238

The Midpoint Formula

If 1 1,L x y and 2 2

,N x y are the endpoints of a segment and M is the

midpoint, then the coordinates of M = 1 2 1 2,2 2

x x y y

. This is also referred

to as the Midpoint Formula.

Example: The coordinates of the endpoints of LG are 23 , and

(8, 9), respectively. What are the coordinates of its midpoint M?

Solution: Let 31 x , 21 y , 82 x , and 92 y . Substitute

these values into the formula 1 2 1 2,2 2

x x y yM

.

3 8 2 9

,2 2

M

or 5 7

,2 2

M

The coordinates of the midpoint of LG are 5 7

,2 2

.

x

y

1 1,L x y

2 2,N x y

1 2 1 2,2 2

x x y yM


DEPED COPY

239

Using the Distance Formula in Proving Geometric Properties Many geometric properties can be proven by using a coordinate plane.

A proof that uses figures on a coordinate plane to prove geometric properties is called a coordinate proof.

To prove geometric properties using the methods of coordinate

geometry, consider the following guidelines for placing figures on a coordinate plane.

1. Use the origin as vertex or center of a figure. 2. Place at least one side of a polygon on an axis. 3. If possible, keep the figure within the first quadrant. 4. Use coordinates that make computations simple and easy.

Sometimes, using coordinates that are multiples of two would make the computation easier.

In some coordinate proofs, the Distance Formula is applied. Example: Prove that the diagonals of a rectangle are

congruent using the methods of coordinate geometry.

Solution:

Given: ABCD with diagonals AC and BD

Prove: BDAC

To prove:

1. Place ABCD on a coordinate plane.

D C

B A

D

C B

A


DEPED COPY

240

2. Label the coordinates as shown below.

a. Find the distance between A and C.

Given: A(0,0) and C(a, b)

2 2

0 0AC a b

2 2AC a b

b. Find the distance between B and D.

Given: B(0, b) and D(a, 0)

Since 2 2AC a b and 2 2BD a b , then

BDAC by substitution.

Therefore, BDAC . The diagonals of a rectangle are

congruent.

D(a, 0)

C(a, b) B(0, b)

A(0, 0)

2 2

0 0BD a b

2 2BD a b


DEPED COPY

241

Learn more about the Distance Formula, the Midpoint Formula, and the Coordinate Proof through the WEB. You may open the following links.

http://www.regentsprep.org/Regents/math/ geometry/GCG2/indexGCG2.htm

http://www.cliffsnotes.com/math/geometry/ coordinate-geometry/midpoint-formula


http://www.cliffsnotes.com/math/geometry/ coordinate-geometry/distance-formula


Your goal in this section is to apply the key concepts of the distance

formula including the midpoint formula and the coordinate proof. Use the mathematical ideas and the examples presented in the preceding section to perform the given activities.

Find the distance between each pair of points on the coordinate plane. Answer the questions that follow.

1. M(2, –3) and N(10, –3) 6. C(–3, 2) and D(9, 7)

2. P(3, –7) and Q(3, 8) 7. S(–4, –2) and T(1, 7)

3. C(–4, 3) and D(7, 6) 8. K(3, –3) and L(–3, 7)

4. A(2, 3) and B(14, 8) 9. E(7, 1) and F(–6, 5)

5. X(–3, 9) and Y(2, 5) 10. R(4, 7) and S(–6, –1)

Questions:

a. How do you find the distance between points that are aligned horizontally? vertically?

b. If two points are not aligned horizontally or vertically, how would you determine the distance between them?

Activity 5:


http://www.regentsprep.org/Regents/math/%20geometry/GCG2/indexGCG2.htm


http://www.cliffsnotes.com/math/geometry/%20coordinate-geometry/midpoint-formula

http://www.cliffsnotes.com/math/geometry/%20coordinate-geometry/midpoint-formula



http://www.cliffsnotes.com/math/geometry/%20coordinate-geometry/distance-formula

http://www.cliffsnotes.com/math/geometry/%20coordinate-geometry/distance-formula

http://www.regentsprep.org/Regents/math/%20%20geometry/GCG4/indexGCG4.htm

http://www.regentsprep.org/Regents/math/%20%20geometry/GCG4/indexGCG4.htm

DEPED COPY

242

Were you able to use the distance formula in finding the distance between each pair of points on the coordinate plane? In the next activity, you will be using the midpoint formula in determining the coordinates of the midpoint of the segment whose endpoints are given.

Find the coordinates of the midpoint of the segment whose endpoints are given below. Explain how you arrived at your answers.

1. A(6, 8) and B(12,10) 6. M(–9, 15) and N(–7, 3)

2. C(5, 11) and D(9, 5) 7. Q(0, 8) and R(–10, 0)

3. K(–3, 2) and L(11, 6) 8. D(12, 5) and E(3, 10)

4. R(–2, 8) and S(10, –6) 9. X(–7, 11) and Y(–9, 3)

5. P(–5, –1) and Q(8, 6) 10. P(–3, 10) and T(–7, –2)

Was it easy for you to determine the coordinates of the midpoint of each segment? I am sure it was. You need this skill in proving geometric relationships using coordinate proof, and in solving real-life problems involving the use of the midpoint formula.

Plot each set of points on the coordinate plane. Then connect the consecutive points by a line segment to form the figure. Answer the questions that follow.

1. A(6, 11), B(1, 2), C(11, 2) 6. L(–4, 4), O(3, 9), V(8, 2), E(1, –3)

2. G(5, 14), O(–3, 8), T(17, –2) 7. S(–1, 5), O(9, –1), N(6, –6), G(–4, 0)

3. F(–2, 6), U(–2, –3), N(7, 6) 8. W(–2, 6), I(9, 6), N(11, –2),

D(–4, –2)

4. L(–2, 8), I(5, 8), K(5, 1), E(–2, 1) 9. B(1, 6), E(13, 7), A(7, –2), T(–5, –3)

5. D(–4, 6), A(8, 6), T(8, –2), 10. C(4, 12), A(9, 9), R(7, 4), E(1, 4),

E(–4, –2) S(–1, –9)

Activity 7:

Activity 6:


DEPED COPY

243

Questions:

a. How do you describe each figure formed? Which figure is a triangle? quadrilateral? pentagon?

b. Which among the triangles formed is isosceles? right?

c. How do you know that the triangle is isosceles? right?

d. Which among the quadrilaterals formed is a square? rectangle? parallelogram? trapezoid?

e. How do you know that the quadrilateral formed is a square? rectangle? parallelogram? trapezoid?

Did you find the activity interesting? Were you able to identify and

describe each figure? In the next activity, you will be using the different properties of geometric figures in determining the missing coordinates.

Name the missing coordinates in terms of the given variables. Answer the questions that follow. 1. COME is a parallelogram. . ∆RST is a right triangle with right

RTS . V is the midpoint of RS .

Activity 8:

y

x E(0, 0)

C(b, c)

M(a, 0)

O(?, ?) y

x T(0, 0)

R(0, 2b)

S(2a, 0)

V(?, ?)


DEPED COPY

244

3. ∆MTC is an isosceles triangle 4. WISE is an isosceles trapezoid. and V is the midpoint of .CT

5. ABCDEF is a regular hexagon. 6. TOPS is a square.

Questions:

a. How did you determine the missing coordinates in each figure?

b. Which guided you in determining the missing coordinates in each figure?

c. In which figure are the missing coordinates difficult to determine? Why?

d. Compare your answers with those of your classmates. Do you have the same answers? Explain.

x

y O(0, d)

P(?, ?)

S(?, ?)

T(–a, b)

x

y

F(a, 0)

E(?, ?)

D(?, ?) C(–a, d)

B(–b, c)

A(?, ?)

y

x E(-a, 0)

W(?, ?)

S(a, 0)

I(b, c)

y

x C(0, 0)

M(?, b)

T(6a, 0) V(?, ?)


DEPED COPY

245

How was the activity you have just done? Was it easy for you to determine the missing coordinates? It was easy for sure!

In this section, the discussion was about the distance formula, the midpoint formula, and the use of coordinate proof.

Now that you know the important ideas about this topic, you can

now move on to the next section and deepen your understanding of these concepts.

Your goal in this section is to think deeper and test further your

understanding of the distance formula and the midpoint formula. You will also write proofs using coordinate geometry. After doing the following activities, you should be able to answer this important question: How does the distance formula facilitate finding solutions to real-life problems and making wise decisions.

Answer the following. 1. The coordinates of the endpoints of ST are (-2, 3) and (3, y), respectively.

Suppose the distance between S and T is 13 units. What value/s of y would satisfy the given condition? Justify your answer.

2. The length of 15MN units. Suppose the coordinates of M are (9, –7) and the coordinates of N are (x, 2).

a. What is the value of x if N lies on the first quadrant? second quadrant?

Explain your answer.

b. What are the coordinates of the midpoint of MN if N lies in the second quadrant? Explain your answer.

3. The midpoint of CS has coordinates (2, –1). If the coordinates of C are

(11, 2), what are the coordinates of S? Explain your answer.

Activity 9:


DEPED COPY

246

4. A tracking device attached to a kidnap victim prior to his abduction indicates that he is located at a point whose coordinates are (8, 10). In the tracking device, each unit on the grid is equivalent to 10 kilometers. How far is the tracker from the kidnap victim if he is located at a point whose coordinates are (1, 3)?

5. The diagram below shows the coordinates of the location of the houses of Luisa and Grace.

Luisa says that the distance of her house from Grace’s house can

be determined by evaluating the expression 2241711 . Grace

does not agree with Luisa. She says that the expression

2214117 gives the distance between their houses. Who do

you think is correct? Justify your answer.

6. A study shed will be constructed midway between two school buildings. On a school map drawn on a coordinate plane, the coordinates of the first building are (10, 30) and the coordinates of the second building are (170, 110).

a. Why do you think the study shed will be constructed midway between the two school buildings?

b. What are the coordinates of the point where the study shed will be constructed?

c. If each unit on the coordinate plane is equivalent to 2 m, what is the distance between the two buildings?

How far would the study shed be from the first building? second building? Explain your answer.

Luisa

Grace

(-7, 4)

(11, 1)

y

x


DEPED COPY

247

x

y

C(a, 0) A(–a, 0)

B(0, a)

7. A Global Positioning System (GPS) device shows that car A travelling at a speed of 60 kph is located at a point whose coordinates are (100, 90). Behind car A is car B, travelling in the same direction at a speed of 80 kph, that is located at a point whose coordinates are (20, 30).

a. What is the distance between the two cars?

b. After how many hours will the two cars be at the same point?

8. Carmela claims that the triangle on the coordinate plane shown on the right is an equilateral triangle. Do you agree with Carmela? Justify your answer.

9. d,aF , d,cA , b,cS , and b,aT are distinct points on the coordinate

plane.

a. Is ATFS ? Justify your answer.

b. What figure will be formed when you connect consecutive points by a line segment? Describe the figure.

y

x

Car B

Car A


DEPED COPY

248

E M

P

R

H

Q

O

S

How was the activity you have just performed? Did you gain better understanding of the lesson? Were you able to use the mathematics concepts learned in solving problems? Were you able to realize the importance of the lesson in the real world? I am sure you were! In the next activity you will be using the distance formula and the coordinate proof in proving geometric relationships.

Write a coordinate proof to prove each of the following. 1. The diagonals of an isosceles trapezoid are congruent.

Given: Trapezoid PQRS with QRPS Prove: QSPR

2. The median to the hypotenuse of a right triangle is half the hypotenuse. Given: ∆LGC is a right triangle with rt. LCG and M is the midpoint of LG .

Prove: LGMC21

3. The segments joining the midpoints of consecutive sides of an isosceles trapezoid form a rhombus. Given: Isosceles trapezoid HOME with OMHE P, Q, R, and S are the midpoints of the sides of the trapezoid. Prove: Quadrilateral PQRS is a rhombus.

Activity 10:

S

P

R

Q

C

L

G

M


DEPED COPY

249

C

A

T

B

S

D

A B

C

C

L

M E G

4. The medians to the legs of an isosceles triangle are congruent.

Given: Isosceles triangle ABC with .AB AC

BT and CS are the medians.

Prove: CSBT 5. If the diagonals of a parallelogram are congruent, then it is a rectangle.

Given: Parallelogram ABCD BDAC Prove: Parallelogram ABCD is a rectangle.

6. If a line segment joins the midpoints of two sides of a triangle, then its length is equal to one-half the length of the third side. Given: Triangle LME C and G are midpoints of

LM and EM , respectively.

Prove: LECG2

1

In this section, the discussion was about the applications of the distance formula, the midpoint formula, and the use of coordinate proofs.

What new realizations do you have about the distance formula, the midpoint formula, and the coordinate proof? In what situations can you use the formulas discussed in this section?

Now that you have a deeper understanding of the topic, you are

ready to do the tasks in the next section.


DEPED COPY

250


situations. You will be given a practical task which will demonstrate your understanding of the distance formula, the midpoint formula, and the use of coordinate proofs.

Perform the following activities. Use the rubric provided to rate your work.

1. Have a copy of the map of your municipality, city, or province then make a sketch of it on a coordinate plane. Indicate on the sketch some important landmarks, then determine their coordinates. Explain why the landmarks you have indicated are significant in your community. Write also a paragraph explaining how you selected the coordinates of these important landmarks.

2. Using the coordinates assigned to the different landmarks in item #1, formulate then solve problems involving the distance formula, midpoint formula, and the coordinate proof.

Rubric for the Sketch of a Map

Score Descriptors 4 The sketch of the map is accurately made, presentable, and

appropriate.

3 The sketch of the map is accurately made and appropriate but not presentable.

2 The sketch of the map is not accurately made but appropriate.

1 The sketch of the map is not accurately made and not appropriate.

Rubric for the Explanation of the Significance of the Landmarks

Score Descriptors 4 The explanations are clear and coherent and the significance of

all the landmarks are justified.

3 The explanations are clear and coherent but the significance of the landmarks are not well justified.

2 The explanations are not so clear and coherent and the significance of the landmarks are not well justified.

1 The explanations are not clear and coherent and the significance of the landmarks are not justified.

Activity 11:


DEPED COPY

251


Score Descriptors

6

Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate

5

Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes

4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes

3

Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details

2 Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension


Source: D.O. #73, s. 2012

In this section, your task was to make a sketch of a map on a coordinate plane and determine the coordinates of some important landmarks. Then using the coordinates assigned to the different landmarks, you were asked to formulate, then, solve problems involving the distance formula and the midpoint formula.


SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about the distance formula, the midpoint formula, and

coordinate proofs and their applications in real life. The lesson provided you with opportunities to find the distance between two points or places, prove geometric relationships using the distance formula, and formulate and solve real-life problems. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the next lesson, Equation of a Circle.


DEPED COPY

252

Start Lesson 2 of this module by relating and connecting previously

learned mathematical concepts to the new lesson, the equation of a circle. As you go through this lesson, think of this important question: “How does the equation of a circle facilitate finding solutions to real-life problems and making wise decisions?” To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier. You may check your work with your teacher.

Determine the number that must be added to make each of the following a perfect square trinomial. Then, express each as a square of a binomial. Answer the questions that follow.

1. 2 4x x _________ 6. 2 9w w _________

2. 2 10t t _________ 7. 2 11x x _________

3. 2 14r r _________ 8. 2 25v v _________

4. 2 22r r _________ 9. 2 13

s s _________

5. 2 36x x _________ 10. 2 34

t t _________

Questions:

a. How did you determine the number that must be added to each expression to produce a perfect square trinomial?

b. How did you express each resulting perfect square trinomial as a square of a binomial?

c. Suppose you are given a square of a binomial. How will you express it as a perfect square trinomial? Give 3 examples.

Activity 1:


DEPED COPY

253

Was it easy for you to determine the number that must be added to the given terms to make each a perfect square trinomial? Were you able to express a perfect square trinomial as a square of a binomial and vice-versa? Completing the square is a prerequisite to your lesson, Equation of a Circle. Do you know why? Find this out as you go through the lesson.

Use the situation below to answer the questions that follow. An air traffic controller (the person who tells the pilot where a plane needs to go using coordinates on the grid) reported that the airport is experiencing air traffic due to the big number of flights that are scheduled to arrive. He advised the pilot of one of the airplanes to move around the airport for the meantime to give way to the other planes to land first. The air traffic controller further told the pilot to maintain its present altitude or height from the ground and its horizontal distance from the origin, point P(0, 0).

Airplane

Air Traffic Controller

Activity 2:


DEPED COPY

254

1. Suppose the plane is located at a point whose coordinates are (30, 40) and each unit on the air traffic controller’s grid is equivalent to 1 km. How far is the plane from the air traffic controller? Explain your answer.

2. What would be the y-coordinate of the position of the plane at a particular instance if its x-coordinate is 5? 10? 15? -20? -30? Explain your answer.

3. Suppose that the pilot strictly follows the advice of the air traffic controller. Is it possible for the plane to be at a point whose x-coordinate is 60? Why?

4. How would you describe the path of the plane as it goes around the airport? What equation do you think would define this path?

Were you able to describe the path of the plane and its location as it

goes around the air traffic controller’s position? Were you able to determine the equation defining the path? How is the given situation related to the new lesson? You will find this out as you go through this lesson.

Perform the following activities. Answer the questions that follow. A. On the coordinate plane below, use a compass to draw a circle with center

at the origin and which passes through A(8, 0).

y

x

Activity 3:


DEPED COPY

255

1. How far is point A from the center of the circle? Explain how you arrived at your answer.

2. Does the circle pass through (0, 8)? How about through (–8, 0)? (0, –8)? Explain your answer.

3. Suppose another point M(–4, 6) is on the coordinate plane. Is M a point on the circle? Why? How about N(9, –2)? Explain your answer.

4. What is the radius of the circle? Explain how you arrived at your answer.

5. If a point is on the circle, how is its distance from the center related to the radius of the circle?

6. How will you find the radius of the circle whose center is at the origin?

B. On the coordinate plane below, use a compass to draw a circle with center

at (3, 1) and which passes through C(9, –4).

1. How far is point C from the center of the circle? Explain how you arrived at your answer.

2. Does the circle pass through (–2, 7)? How about through (8, 7)? (–3, –4)? Explain your answer.

3. Suppose another point M(–7, 6) is on the coordinate plane. Is M a point on the circle? Why?

4. What is the radius of the circle? Explain how you arrived at your answer.

5. How will you find the radius of the circle whose center is not at the origin?

x

y


DEPED COPY

256

Were you able to determine if a circle passes through a given point? Were you able to find the radius of a circle given the center? What equation do you think would relate the radius and the center of a circle? Find this out as you go through the lesson.

How did you find the preceding activities? Are you ready to learn about the equation of a circle? I am sure you are!

From the activities you have done, you were able to find the square of a binomial, a mathematics skill that is needed in understanding the equation of a circle. You were also able to find out how circles are illustrated in real life. You were also given the opportunity to find the radius of a circle and determine if a point is on the circle or not. But how does the equation of a circle help in solving real-life problems and in making wise decisions? You will find these out in the succeeding activities. Before doing these activities, read and understand first some important notes on the equation of a circle and the examples presented.

The Standard Form of the Equation of a Circle

The standard equation of a circle with center at (h, k) and a radius of r

units is 2 2 2x h y k r . The values of h and k indicate that the circle

is translated h units horizontally and k units vertically from the origin. If the center of the circle is at the origin, the equation of the circle is

2 2 2x y r .

x (h,k)

(0,0)

Circle with center at (h, k) Circle with center at the origin

2 2 2x h y k r

2 2 2x y r

x

y,xQ y,xP

r r

y y


DEPED COPY

257

Example 1: The equation of a circle with center at (2, 7) and a radius

of 6 units is 2 2 22 7 6x y or

2 2

2 7 36x y .

Example 2: The equation of a circle with center at (–5, 3) and a

radius of 12 units is 2 2 25 3 12x y or

2 2

5 3 144x y .

Example 3: The equation of a circle with center at (–4, –9) and a

radius of 8 units is 2 2 24 9 8x y or

2 2

4 9 64x y .

Example 4: The equation of a circle with center at the origin and a

radius of 4 units is 2 2 24x y or 2 2 16x y .

Example 5: The equation of a circle with center at the origin and a

radius of 15 units is 2 2 215x y or 2 2 225x y .

The General Equation of a Circle

The general equation of a circle is 2 2 0x y Dx Ey F , where D,

E, and F are real numbers. This equation is obtained by expanding the

standard equation of a circle, 2 2 2x h y k r .

2 2 2x h y k r 2 2 2 2 22 2x hx h y ky k r

2 2 2 2 22 2x hx h y ky k r

2 2 2 2 22 2x y hx ky h k r

2 2 2 2 22 2 0x y hx ky h k r

If 2 ,D h 2 ,E k and 2 2 2,F h k r the equation

2 2 2 2 22 2 0x y hx ky h k r becomes 2 2 0.x y Dx Ey F


DEPED COPY

258

Example: Write the general equation of a circle with center C(4, –1) and a radius of 7 units. Then determine the values of D, E, and F.

The center of the circle is at (h, k), where h = 4 and k = –1. Substitute these values in the standard form of the equation of a acircle together with the length of the radius r which is equal to 7 units.

2 2 2x h y k r

2 2 24 1 7x y

Simplify 2 2 24 1 7x y .

2 2 24 1 7x y 2 28 16 2 1 49x x y y

2 28 16 2 1 49x x y y

2 2 8 2 17 49x y x y

2 2 8 2 17 49 0x y x y

2 2 8 2 32 0x y x y

Answer: 2 2 8 2 32 0x y x y is the general equation

of the circle with center C(4, –1) and radius of 7 units. In the equation, D = –8, E = 2, and F = –32.

Finding the Center and the Radius of a Circle Given the Equation The center and the radius of a circle can be found given the equation. To do this, transform the given equation to its standard form

2 2 2x h y k r if the center of the circle is k,h , or 2 2 2x y r if

the center of the circle is the origin. Once the center and the radius of the circle are found, its graph can be shown on the coordinate plane.

Example 1: Find the center and the radius of the circle 2 2 64,x y

and then draw its graph.


DEPED COPY

259

Solution: The equation of the circle 2 2 64x y has its center at

the origin. Hence, it can be transformed to the form.2 2 2x y r . 2 2 64x y 2 2 28x y

The center of the circle is (0, 0) and its radius is 8 units. Its graph is shown below.

Example 2: Determine the center and the radius of the circle

2 2

2 4 25,x y and draw its graph.

Solution: The equation of the circle 2 2

2 4 25x y can be

written in the form 2 2 2x h y k r .

2 2

2 4 25x y 2 2 22 4 5x y

The center of the circle is (2, 4) and its radius is 5 units.

Its graph is shown below.

x

y

r = 8

x

y

r = 5


DEPED COPY

260

Example 3: What is the center and the radius of the circle 2 2 6 10 18 0x y x y ? Show the graph.

Solution: The equation of the circle 2 2 6 10 18 0x y x y is

written in general form. To determine its center and radius, write the equation in the form

2 2 2x h y k r .

2 2 6 10 18 0x y x y 2 26 10 18x x y y

Add to both sides of the equation 2 26 10 18x x y y the square of one-half the

coefficient of x and the square of one-half the coefficient of y.

Simplify 2 26 9 10 25 18 9 25x x y y .

2 26 9 10 25 16x x y y

2 26 9 10 25 16x x y y

Rewriting, we obtain 2 2

3 5 16x y or

2 2 23 5 4x y

The center of the circle is at (3, 5) and its radius is 4 units.

1

6 32

; 2

3 9 5102

1 ;

2

5 25

x

y

r = 4


DEPED COPY

261

Example 4: What is the center and the radius of the circle 4x2 + 4y2 + 12x – 4y – 90 = 0? Show the graph.

Solution: 4x2 + 4y2 + 12x – 4y – 90 = 0 is an equation of a circle

that is written in general form. To determine its center and radius, write the equation in the form

2 2 2x h y k r .

2 24 4 12 4 90 0x y x y or 2 24 4 12 4 90x y x y

Divide both sides of the equation by 4.

2 24 4 12 4 90x y x y 2 24 4 12 4 90

4 4

x y x y

2 2 903

4x y x y

Add on both sides of the equation 2 2 903

4x y x y

the square of one-half the coefficient of x and the square of one-half the coefficient of y.

Simplify 2 29 1 90 9 13

4 4 4 4 4x x y y .

100

4

2 29 13 25

4 4x x y y

Rewriting, we have

2 2

3 125

2 2x y

.

1 3

32 2

;

2

3 9

2 4

1 11

2 2 ;

2

1 1

2 4


DEPED COPY

262

Write the equation 2 23 1 25

2 2x y

in the form

2 2 2,x h y k r that is

2 223 1 5

2 2x y

The center of the circle is at 3 1, 2 2

and its radius is

5 units.

Learn more about the Equation of a Circle through the WEB. You may open the following links.

http://www.mathopenref.com/coordbasiccircle.html

http://www.mathopenref.com/coordgeneralcircle.html

https://www.khanacademy.org/math/geometry/cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem

http://www.math-worksheet.org/using-equations-of-circles

x

y

r = 5





DEPED COPY

263

Your goal in this section is to apply the key concepts of the equation

of a circle. Use the mathematical ideas and the examples presented in the preceding section to perform the activities that follow.

Determine the center and the radius of the circle that is defined by each of the following equations. Then graph each circle on a coordinate plane (or use GeoGebra to graph each). Answer the questions that follow. 1. 2 2 49x y 2.

2 25 6 81x y

Activity 4:

x

y

x

y


DEPED COPY

264

3. 2 2 100x y 4. 2 2

7 1 49x y

5. 2 2 8 6 39 0x y x y 6. 2 2 10 16 32 0x y x y

\ Questions:

a. How did you determine the center of each circle? How about the radius?

b. How do you graph circles that are defined by equations of the form

2 2 2x y r ? 2 2 2x h y k r ?

2 2 0x y Dx Ey F ?

y

x

y

x x

x

y

y


DEPED COPY

265

How was the activity? Did it challenge you? Were you able to determine the center and the radius of the circle? I am sure you were! In the next activity, you will write the equation of the circle as described.

Write the equation of each of the following circles given the center and the radius. Answer the questions that follow.

Center Radius 1. origin 12 units 2. (2, 6) 9 units 3. (–7, 2) 15 units 4. (–4, –5) 5 2 units 5. (10, –8) 3 3 units

Questions:

a. How do you write the equation of a circle, given its radius, if the center is at the origin?

b. How about if the center is not at (0, 0)? c. Suppose two circles have the same center. Should the equations

defining these circles be the same? Why?

Were you able to write the equation of the circle given its radius and its center? I know you were! In the next activity, you will write the equation of a circle from standard to general form.

Write each equation of a circle in general form. Show your solutions completely.

1. 2 22 4 36x y 6.

2 27 64x y

2. 2 24 9 144x y 7.

22 2 49x y

3. 2 26 1 81x y 8.

2 22 100x y

4. 2 28 7 225x y 9.

2 25 5 27x y

5. 22 5 36x y 10.

2 24 4 32x y

Activity 6:

Activity 5:


DEPED COPY

266

How did you find the activity? Were you able to write all the equations in their general form? Did the mathematics concepts and principles that you previously learned help you in transforming the equations? In the next activity, you will do the reverse. This time, you will transform the equation of a circle from general to standard form, then determine the radius and the center of the circle.

In numbers 1 to 6, a general equation of a circle is given. Transform the equation to standard form, then give the coordinates of the center and the radius. Answer the questions that follow.

1. 2 2 2 8 47 0x y x y 4. 2 2 8 84 0x y y 2. 2 2 4 4 28 0x y x y 5. 2 29 9 12 6 31 0x y x y 3. 2 2 10 4 3 0x y x y 6. 2 24 4 20 12 2 0x y x y

Questions:

a. How did you write each general equation of a circle to standard form?

b. What mathematics concepts or principles did you apply in transforming each equation to standard form?

c. Is there a shorter way of transforming each equation to standard form? Describe this way, if there is any.

Were you able to write each equation of a circle from general form

to standard form? Were you able find a shorter way of transforming each equation to standard form?

In this section, the discussion was about the equation of a circle, its radius and center, and the process of transforming the equation from one form to another.


Now that you know the important ideas about this topic, let us

deepen your understanding by moving on to the next section.

Activity 7:


DEPED COPY

267

Your goal in this section is to test further your understanding of the

equation of a circle by solving more challenging problems involving this concept. After doing the following activities, you should be able to find out how the equations of circles are used in solving real-life problems and in making decisions.

Determine which of the following equations describe a circle and which do not. Justify your answer.

1. 2 2 2 8 26 0x y x y 3. 2 2 6 8 32 0x y x y 2. 2 2 9 4 10x y x y 4. 2 2 8 14 65 0x y x y

How was the activity? Were you able to determine which are circles

and which are not? In the next activity, you will further deepen your understanding about the equation of a circle and solve real-life problems.

Answer the following.

1. The diameter of a circle is 18 units and its center is at (–3, 8). What is the equation of the circle?

2. Write an equation of the circle with a radius of 6 units and is tangent to the line 1y at (10, 1).

3. A circle defined by the equation 2 26 9 34x y is tangent to a

line at the point (9, 4). What is the equation of the line?

4. A line passes through the center of a circle and intersects it at points (2, 3) and (8, 7). What is the equation of the circle?

5. The Provincial Disaster and Risk Reduction Management Committee (PDRRMC) advised the residents living within the 10 km radius critical area to evacuate due to eminent eruption of a volcano. On the map that is drawn on a coordinate plane, the coordinates corresponding to the location of the volcano is (3, 4).

Activity 9:

Activity 8:


DEPED COPY

268

a. If each unit on the coordinate plane is equivalent to 1 km, what is the equation of the circle enclosing the critical area?

b. Suppose you live at point (11, 6). Would you follow the advice of the PDRRMC? Why?

c. In times of eminent disaster, what precautionary measures should you take to be safe?

d. Suppose you are the leader of a two-way radio team with 15 members that is tasked to give warnings to the residents living within the critical area. Where would you position each member of the team who is tasked to inform the other members as regards the current situation and to warn the residents living within his/her assigned area? Explain your answer.

6. Cellular phone networks use towers to transmit calls to a circular area. On a grid of a province, the coordinates that correspond to the location of the towers and the radius each covers are as follows: Wise Tower is at (–5, –3) and covers a 9 km radius; Global Tower is at (3, 6) and covers a 4 km radius; and Star Tower is at (12, –3) and covers a 6 km radius. a. What equation represents the transmission boundaries of each

tower? b. Which tower transmits calls to phones located at (12, 2)? (–6, –7)?

(2, 8)? (1, 3)? c. If you were a cellular phone user, which cellular phone network will

you subscribe to? Why?

Did you find the activity challenging? Were you able to answer all the questions and problems involving the equations of circles? I am sure you were!

In this section, the discussion was about your understanding of the equation of a circle and their applications in real life.

What new realizations do you have about the equation of a circle? How would you connect this to real life? How would you use this in making wise decisions?



DEPED COPY

269

Your goal in this section is to apply your learning to real-life situations.

You will be given a practical task which will demonstrate your understanding of the equation of a circle.

On a clean sheet of grid paper, paste some small pictures of objects such that they are positioned at different coordinates. Then, draw circles that contain these pictures. Using the pictures and the circles drawn on the grid, formulate and solve problems involving the equation of the circle, then solve them. Use the rubric provided to rate your work.

Rubric for a Scrapbook Page

Score Descriptors 4 The scrapbook page is accurately made, presentable, and appropriate. 3 The scrapbook page is accurately made and appropriate. 2 The scrapbook page is not accurately made but appropriate. 1 The scrapbook page is not accurately made and not appropriate.


Score Descriptors

6




3




Source: D.O. #73, s. 2012

Activity 10:


DEPED COPY

270

In this section, your task was to formulate problems involving the equation of a circle using the pictures of objects that you positioned on a grid.



This lesson was about the equations of circles and their applications in real life. The lesson provided you with opportunities to give the equations of circles and use them in practical situations. Moreover, you were given the chance to formulate and solve real-life problems. Understanding this lesson and relating it to the mathematics concepts and principles that you have previously learned is essential in any further work in mathematics. GLOSSARY OF TERMS Coordinate Proof – a proof that uses figures on a coordinate plane to prove geometric relationships Distance Formula – an equation that can be used to find the distance between any pair of points on the coordinate plane. The distance formula is

2 2

2 1 2 1d x x y y or

2 2

2 1 2 1,PQ x x y y if 1 1

, P x y

and 2 2, Q x y are points on a coordinate plane.

Horizontal Distance (between two points) – the absolute value of the difference of the x-coordinates of two points

Midpoint – a point on a line segment and divides the same segment into two equal parts

Midpoint Formula – a formula that can be used to find the coordinates of the midpoint of a line segment on the coordinate plane. The midpoint of

1 1, P x y and 2 2

, Q x y is 1 2 1 2,2 2

x x y y

.

The General Equation of a Circle – the equation of a circle obtained by

expanding 2 2 2x h y k r . The general equation of a circle is

2 2 0x y Dx Ey F , where D, E, and F are real numbers.


DEPED COPY

271

The Standard Equation of a Circle – the equation that defines a circle with

center at (h, k) and a radius of r units. It is given by 2 2 2x h y k r .

Vertical Distance (between two points) – the absolute value of the difference of the y-coordinates of two points DepEd Instructional Materials that can be used as additional resources: 1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third

Year Mathematics. Plane Coordinate Geometry. Module 20: Distance and Midpoint Formulae

2. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third

Year Mathematics. Plane Coordinate Geometry. Module 22: Equation of a Circle

3. Distance Learning Module (DLM) 3, Module 3: Plane Coordinate

Geometry.

4. EASE Modules Year III, Module 2: Plane Coordinate Geometry REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: References: Bass, Laurie E., Randall I. Charles, Basia Hall, Art Johnson, and Dan

Kennedy. Texas Geometry. Pearson Prentice Hall, Boston, Massachusetts 02116, 2008.

Bass, Laurie E., Basia Rinesmith Hall, Art Johnson, and Dorothy F. Wood.

Prentice Hall Geometry Tools for a Changing World. Prentice-Hall, Inc., NJ, USA, 1998.

Boyd, Cummins, Malloy, Carter, and Flores. Glencoe McGraw-Hill Geometry.

The McGraw-Hill Companies, Inc., USA, 2008. Callanta, Melvin M. Infinity, Worktext in Mathematics III. EUREKA Scholastic

Publishing, Inc., Makati City, 2012. Chapin, Illingworth, Landau, Masingila and McCracken. Prentice Hall Middle

Grades Math, Tools for Success, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1997.


DEPED COPY

272

Cifarelli, Victor, et al. cK-12 Geometry, Flexbook Next Generation Textbooks, Creative Commons Attribution-Share Alike, USA, 2009.

Clemens, Stanley R., Phares G. O’Daffer, Thomas J. Cooney, and John A.

Dossey. Addison-Wesley Geometry. Addison-Wesley Publishing Company, Inc., USA, 1990.

Clements, Douglas H., Kenneth W. Jones, Lois Gordon Moseley, and Linda

Schulman. Math in my World, McGraw-Hill Division, Farmington, New York, 1999.

Department of Education. K to 12 Curriculum Guide Mathematics,

Department of Education, Philippines, 2012. Gantert, Ann Xavier. AMSCO’s Geometry. AMSCO School Publications, Inc.,

NY, USA, 2008. Renfro, Freddie L. Addison-Wesley Geometry Teacher’s Edition. Addison-

Wesley Publishing Company, Inc., USA, 1992. Rich, Barnett and Christopher Thomas. Schaum’s Outlines Geometry Fourth

Edition. The McGraw-Hill Companies, Inc., USA, 2009. Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and

Marvin L. Bittinger. Addison-Wesley Informal Geometry. Addison-Wesley Publishing Company, Inc., USA, 1992.

Wilson, Patricia S., et al. Mathematics, Applications and Connections, Course

I, Glencoe Division of Macmillan/McGraw-Hill Publishing Company, Westerville, Ohio, 1993.

Website Links as References and Sources of Learning Activities: CliffsNotes. Midpoint Formula. (2013). Retrieved June 29, 2014, from http://www.cliffsnotes.com/math/geometry/coordinate-geometry/midpoint-formula CliffsNotes. Distance Formula. (2013). Retrieved June 29, 2014, from http://www.cliffsnotes.com/math/geometry/coordinate-geometry/distance-formula Math Open Reference. Basic Equation of a Circle (Center at 0,0). (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/ coordbasiccircle.html


http://www.mathopenref.com/

DEPED COPY

273

Math Open Reference. Equation of a Circle, General Form (Center anywhere). (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/coordgeneralcircle.html Math-worksheet.org. Using equations of circles. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/using-equations-of-circles Math-worksheet.org. Writing equations of circles. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/writing-equations-of-circles Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved June 29, 2014, from http://www.regentsprep.org/Regents/ math/geometry/GCG2/ Lmidpoint.htm Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved June 29, 2014, from http://www.regentsprep.org/Regents/math/geometry/GCG3/ Ldistance.htm Stapel, Elizabeth. "Conics: Circles: Introduction & Drawing." Purplemath. Retrieved June 29, 2014, from http://www.purplemath.com/modules/ circle.htm

Website Links for Videos: Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved June 29, 2014, from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem Khan Academy. Completing the square to write equation in standard form of a circle. Retrieved June 29, 2014, from https://www.khanacademy.org/math/ geometry/cc-geometry-circles/equation-of-a-circle/v/completing-the-square-to-write-equation-in-standard-form-of-a-circle Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved June 29, 2014, from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem


http://www.regentsprep.org/Regents/

http://www.regentsprep.org/Regents/math/geometry/GCG3/

http://www.purplemath.com/modules/

https://www.khanacademy.org/math/geometry/

https://www.khanacademy.org/math/


DEPED COPY

274

Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved June 29, 2014, from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem Ukmathsteacher. Core 1 – Coordinate Geometry (3) – Midpoint and distance formula and Length of Line Segment. Retrieved June 29, 2014, from http://www.youtube.com/watch?v=qTliFzj4wuc VividMaths.com. Distance Formula. Retrieved June 29, 2014, from http://www.youtube.com/watch?v=QPIWrQyeuYw Website Links for Images:

asiatravel.com. Pangasinan Map. Retrieved June 29, 2014, from http://www.asiatravel.com/philippines/pangasinan/pangasinanmap.jpg

DownTheRoad.org. Pictures of, Chengdu to Kangding, China Photo, Images, Picture from. (2005). Retrieved June 29, 2014, from http://www.downtheroad.org/Asia/Photo/9Sichuan_China_Image/3Chengdu_Kangding_China.htm

Hugh Odom Vertical Consultants. eleven40 theme on Genesis Framework · WordPress. Cell Tower Development – How Are Cell Tower Locations Selected? Retrieved June 29, 2014, from http://blog.thebrokerlist.com/cell-tower-development-how-are-cell-tower-locations-selected/

LiveViewGPS, Inc. GPS Tracking PT-10 Series. (2014). Retrieved June 29, 2014, from http://www.liveviewgps.com/gps+tracking+device+pt-10+series.html

Sloan, Chris. Current "1991" Air Traffic Control Tower at Amsterdam Schiphol Airport – 2012. (2012). Retrieved June 29, 2014, from http://airchive.com/html/airplanes-and-airports/amsterdam-schipol-airport-the-netherlands-/current-1991-air-traffic-control-tower-at-amsterdam-schiphol-airport-2012-/25510

wordfromthewell.com. Your Mind is Like an Airplane. (2012). Retrieved June 29, 2014, from http://wordfromthewell.com/2012/11/14/your-mind-is-like-an-airplane/



http://www.studiopress.com/themes/eleven40

http://www.studiopress.com/

http://wordpress.org/

http://airchive.com/html/airplanes-and-airports/amsterdam-schipol-airport-the-netherlands-/current-1991-air-traffic-control-tower-at-amsterdam-schiphol-airport-2012-/25510



mathematics - sweet formula - math resources

Documents