chapter 7 resource masters - math class - home · pdf fileup to twenty of the key vocabulary...

Chapter 7Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 7 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860184-3 GeometryChapter 7 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix

Lesson 7-1Study Guide and Intervention . . . . . . . . 351–352Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 353Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Reading to Learn Mathematics . . . . . . . . . . 355Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 356







Chapter 7 AssessmentChapter 7 Test, Form 1 . . . . . . . . . . . . 393–394Chapter 7 Test, Form 2A . . . . . . . . . . . 395–396Chapter 7 Test, Form 2B . . . . . . . . . . . 397–398Chapter 7 Test, Form 2C . . . . . . . . . . . 399–400Chapter 7 Test, Form 2D . . . . . . . . . . . 401–402Chapter 7 Test, Form 3 . . . . . . . . . . . . 403–404Chapter 7 Open-Ended Assessment . . . . . . 405Chapter 7 Vocabulary Test/Review . . . . . . . 406Chapter 7 Quizzes 1 & 2 . . . . . . . . . . . . . . . 407Chapter 7 Quizzes 3 & 4 . . . . . . . . . . . . . . . 408Chapter 7 Mid-Chapter Test . . . . . . . . . . . . 409Chapter 7 Cumulative Review . . . . . . . . . . . 410Chapter 7 Standardized Test Practice . 411–412Unit 2 Test/Review (Ch. 4–7) . . . . . . . . 413–414First Semester Test (Ch. 1–7) . . . . . . . 415–416

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A34

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 7 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 7 Resource Masters includes the core materials neededfor Chapter 7. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.

WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

© Glencoe/McGraw-Hill v Glencoe Geometry

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

Assessment OptionsThe assessment masters in the Chapter 7Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 398–399. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

77

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

ambiguous case

angle of depression

angle of elevation

cosine

geometric mean

Law of Cosines

Law of Sines

Pythagorean identity

puh·thag·uh·REE·ahn

(continued on the next page)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

Pythagorean triple

reciprocal identity

ri·SIP·ruh·kuhl

sine

solve a triangle

tangent

trigonometric identity

trig·uh·nuh·MET·rik

trigonometric ratio

trigonometry

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

77

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

77

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 7. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 7.1

Theorem 7.2

Theorem 7.3

Theorem 7.4Pythagorean Theorem

Theorem 7.5Converse of the Pythagorean Theorem

Theorem 7.6

Theorem 7.7

Study Guide and InterventionGeometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

© Glencoe/McGraw-Hill 351 Glencoe Geometry

Less

on

7-1

Geometric Mean The geometric mean between two numbers is the square root oftheir product. For two positive numbers a and b, the geometric mean of a and b is the positive number x in the proportion !

ax! " !b

x!. Cross multiplying gives x2 " ab, so x " !ab".

Find the geometric mean between each pair of numbers.

a. 12 and 3Let x represent the geometric mean.

!1x2! " !3

x! Definition of geometric mean

x2 " 36 Cross multiply.

x " !36" or 6 Take the square root of each side.

b. 8 and 4Let x represent the geometric mean.

!8x! " !4

x!

x2 " 32x " !32"

# 5.7

ExercisesExercises

Find the geometric mean between each pair of numbers.

1. 4 and 4 2. 4 and 6

3. 6 and 9 4. !12! and 2

5. 2!3" and 3!3" 6. 4 and 25

7. !3" and !6" 8. 10 and 100

9. !12! and !

14! 10. and

11. 4 and 16 12. 3 and 24

The geometric mean and one extreme are given. Find the other extreme.

13. !24" is the geometric mean between a and b. Find b if a " 2.

14. !12" is the geometric mean between a and b. Find b if a " 3.

Determine whether each statement is always, sometimes, or never true.

15. The geometric mean of two positive numbers is greater than the average of the twonumbers.

16. If the geometric mean of two positive numbers is less than 1, then both of the numbersare less than 1.

3!2"!5

2!2"!5

ExampleExample


Altitude of a Triangle In the diagram, !ABC $ !ADB $ !BDC.An altitude to the hypotenuse of a right triangle forms two right triangles. The two triangles are similar and each is similar to the original triangle. CD

B

A

Study Guide and Intervention (continued)

Geometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

Use right !ABC with B!D! ⊥ A!C!. Describe two geometricmeans.

a. !ADB " !BDC so !BAD

D! " !BC

DD!.

In !ABC, the altitude is the geometricmean between the two segments of thehypotenuse.

b. !ABC " !ADB and !ABC " !BDC,

so !AACB! " !A

ADB! and !B

ACC! " !D

BCC!.

In !ABC, each leg is the geometricmean between the hypotenuse and thesegment of the hypotenuse adjacent tothat leg.

Find x, y, and z.

!PP

QR! " !

PP

QS!

!21

55! " !

1x5! PR " 25, PQ " 15, PS " x

25x " 225 Cross multiply.x " 9 Divide each side by 25.

Theny " PR # SP

" 25 # 9" 16

!QPR

R! " !QR

RS!

!2z5! " !y

z! PR " 25, QR " z, RS " y

!2z5! " !1

z6! y " 16

z2 " 400 Cross multiply.z " 20 Take the square root of each side.

zy

x

15

R

Q P

S25

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find x, y, and z to the nearest tenth.

1. 2. 3.

4. 5. 6.x zy

62

x

z y

2

2xy

1

!%3

!"12

zx y

81

z

xy 5

2

x

1 3

Skills PracticeGeometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1


Less

on

7-1

Find the geometric mean between each pair of numbers. State exact answers andanswers to the nearest tenth.

1. 2 and 8 2. 9 and 36 3. 4 and 7

4. 5 and 10 5. 2!2" and 5!2" 6. 3!5" and 5!5"

Find the measure of each altitude. State exact answers and answers to the nearesttenth.

7. 8.

9. 10.

Find x and y.

11. 12.

13. 14.

2

5y

x15

4

y

x

10

4

yx

3 9

yx

R T

S

U4.5 8G

E H

F

2

9

L

M

N

P 2

12

C

D

B

A 2

7


Find the geometric mean between each pair of numbers to the nearest tenth.

1. 8 and 12 2. 3!7" and 6!7" 3. !45! and 2

Find the measure of each altitude. State exact answers and answers to the nearesttenth.

4. 5.

Find x, y, and z.

6. 7.

8. 9.

10. CIVIL ENGINEERING An airport, a factory, and a shopping center are at the vertices of aright triangle formed by three highways. The airport and factory are 6.0 miles apart. Theirdistances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service roadwill be constructed from the shopping center to the highway that connects the airport andfactory. What is the shortest possible length for the service road? Round to the nearesthundredth.

x y

10z

20x

y

2

3

z

zx y

625

23

z

xy

8

17

6

KL

J M

125

U

T A V

Practice Geometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

Reading to Learn MathematicsGeometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1


Less

on

7-1

Pre-Activity How can the geometric mean be used to view paintings?

Read the introduction to Lesson 7-1 at the top of page 342 in your textbook.

• What is a disadvantage of standing too close to a painting?

• What is a disadvantage of standing too far from a painting?

Reading the Lesson1. In the past, when you have seen the word mean in mathematics, it referred to the

average or arithmetic mean of the two numbers.

a. Complete the following by writing an algebraic expression in each blank.

If a and b are two positive numbers, then the geometric mean between a and b is

and their arithmetic mean is .

b. Explain in words, without using any mathematical symbols, the difference betweenthe geometric mean and the algebraic mean.

2. Let r and s be two positive numbers. In which of the following equations is z equal to thegeometric mean between r and s?

A. !zs

! " !zr! B. !z

r! " !z

s! C. s : z " z: r D. !z

r! " !

zs! E. !

zr! " !

zs! F. !

zs! " !z

r!

3. Supply the missing words or phrases to complete the statement of each theorem.

a. The measure of the altitude drawn from the vertex of the right angle of a right triangle

to its hypotenuse is the between the measures of the two

segments of the .

b. If the altitude is drawn from the vertex of the angle of a right

triangle to its hypotenuse, then the measure of a of the triangle

is the between the measure of the hypotenuse and the segment

of the adjacent to that leg.

c. If the altitude is drawn from the of the right angle of a right

triangle to its , then the two triangles formed are

to the given triangle and to each other.

Helping You Remember4. A good way to remember a new mathematical concept is to relate it to something you

already know. How can the meaning of mean in a proportion help you to remember howto find the geometric mean between two numbers?


Mathematics and MusicPythagoras, a Greek philosopher who lived during the sixth century B.C.,believed that all nature, beauty, and harmony could be expressed by whole-number relationships. Most people remember Pythagoras for his teachingsabout right triangles. (The sum of the squares of the legs equals the square ofthe hypotenuse.) But Pythagoras also discovered relationships between themusical notes of a scale. These relationships can be expressed as ratios.

C D E F G A B C$

!11! !

89! !

45! !

34! !

23! !

35! !1

85! !

12!

When you play a stringed instrument, The C string can be usedyou produce different notes by placing to produce F by placingyour finger on different places on a string. a finger !

34! of the way

This is the result of changing the length along the string.of the vibrating part of the string.

Suppose a C string has a length of 16 inches. Write and solve proportions to determine what length of string would have to vibrate to produce the remaining notes of the scale.

1. D 2. E 3. F

4. G 5. A 6. B

7. C$

8. Complete to show the distance between finger positions on the 16-inch

C string for each note. For example, C(16) # D&14!29!' " 1!

79!.

C D E F G A B C$

9. Between two consecutive musical notes, there is either a whole step or a half step. Using the distances you found in Exercise 8, determine what two pairs of notes have a half step between them.

1!79! in.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

34 of C string

Study Guide and InterventionThe Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

The Pythagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.

!ABC is a right triangle, so a2 % b2 " c2.

Prove the Pythagorean Theorem.With altitude C"D", each leg a and b is a geometric mean between hypotenuse c and the segment of the hypotenuse adjacent to that leg.

!ac

! " !ay! and !b

c! " !

bx!, so a2 " cy and b2 " cx.

Add the two equations and substitute c " y % x to geta2 % b2 " cy % cx " c( y % x) " c2.

c yx a

b

hA C

BD

c a

bA C

B

Example 1Example 1

Example 2Example 2

a. Find a.

a2 % b2 " c2 Pythagorean Theorem

a2 % 122 " 132 b " 12, c " 13

a2 % 144 " 169 Simplify.a2 " 25 Subtract.

a " 5 Take the square root of each side.

a

12

13

AC

B

b. Find c.

a2 % b2 " c2 Pythagorean Theorem

202 % 302 " c2 a " 20, b " 30

400 % 900 " c2 Simplify.

1300 " c2 Add.

!1300" " c Take the square root of each side.

36.1 # c Use a calculator.

c

30

20

AC

B

ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6. x

1128

x

33

16x

59

49

x

6525

x

159

x

3 3


Converse of the Pythagorean Theorem If the sum of the squares of the measures of the two shorter sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

If the three whole numbers a, b, and c satisfy the equation a2 % b2 " c2, then the numbers a, b, and c form a If a2 % b2 " c2, then Pythagorean triple. !ABC is a right triangle.

Determine whether !PQR is a right triangle.a2 % b2 " c2 Pythagorean Theorem

102 % (10!3")2 " 202 a " 10, b " 10!3", c " 20

100 % 300 " 400 Simplify.

400 " 400✓ Add.

The sum of the squares of the two shorter sides equals the square of the longest side, so thetriangle is a right triangle.

Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.

1. 30, 40, 50 2. 20, 30, 40 3. 18, 24, 30

4. 6, 8, 9 5. !37!, !

47!, !

57! 6. 10, 15, 20

7. !5", !12", !13" 8. 2, !8", !12" 9. 9, 40, 41

A family of Pythagorean triples consists of multiples of known triples. For eachPythagorean triple, find two triples in the same family.

10. 3, 4, 5 11. 5, 12, 13 12. 7, 24, 25

10!%3

20 10

QR

P

c

ab

A

C

B


The Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

ExampleExample

ExercisesExercises

Skills PracticeThe Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

Find x.

1. 2. 3.

4. 5. 6.

Determine whether !STU is a right triangle for the given vertices. Explain.

7. S(5, 5), T(7, 3), U(3, 2) 8. S(3, 3), T(5, 5), U(6, 0)

9. S(4, 6), T(9, 1), U(1, 3) 10. S(0, 3), T(#2, 5), U(4, 7)

11. S(#3, 2), T(2, 7), U(#1, 1) 12. S(2, #1), T(5, 4), U(6, #3)


13. 12, 16, 20 14. 16, 30, 32 15. 14, 48, 50

16. !25!, !

45!, !

65! 17. 2!6", 5, 7 18. 2!2", 2!7", 6

x

31

14x9 9

8

x12.5

25

x

1232

x

12

13x

12

9


Find x.

1. 2. 3.

4. 5. 6.

Determine whether !GHI is a right triangle for the given vertices. Explain.

7. G(2, 7), H(3, 6), I(#4, #1) 8. G(#6, 2), H(1, 12), I(#2, 1)

9. G(#2, 1), H(3, #1), I(#4, #4) 10. G(#2, 4), H(4, 1), I(#1, #9)


11. 9, 40, 41 12. 7, 28, 29 13. 24, 32, 40

14. !95!, !

152!, 3 15. !

13!, , 1 16. , , !

47!

17. CONSTRUCTION The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 11 feet long. How high is the dock?

11 ft?

dock

ramp

10 ft

2!3"!7

!4"!7

2!2"!3

x 2424

42

x16

14

x

34

22

x26

2618

x

34 21x

13

23

Practice The Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

Reading to Learn MathematicsThe Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

Pre-Activity How are right triangles used to build suspension bridges?


Do the two right triangles shown in the drawing appear to be similar?Explain your reasoning.

Reading the Lesson

1. Explain in your own words the difference between how the Pythagorean Theorem is usedand how the Converse of the Pythagorean Theorem is used.

2. Refer to the figure. For this figure, which statements are true?

A. m2 % n2 " p2 B. n2 " m2 % p2

C. m2 " n2 % p2 D. m2 " p2 # n2

E. p2 " n2 # m2 F. n2 # p2 " m2

G. n " !m2 %"p2" H. p " !m2 #"n2"

3. Is the following statement true or false?A Pythagorean triple is any group of three numbers for which the sum of the squares of thesmaller two numbers is equal to the square of the largest number. Explain your reasoning.

4. If x, y, and z form a Pythagorean triple and k is a positive integer, which of the followinggroups of numbers are also Pythagorean triples?

A. 3x, 4y, 5z B. 3x, 3y, 3z C. #3x, #3y, #3z D. kx, ky, kz

Helping You Remember

5. Many students who studied geometry long ago remember the Pythagorean Theorem as theequation a2 % b2 " c2, but cannot tell you what this equation means. A formula is uselessif you don’t know what it means and how to use it. How could you help someone who hasforgotten the Pythagorean Theorem remember the meaning of the equation a2 % b2 " c2?

pm

n


Converse of a Right Triangle TheoremYou have learned that the measure of the altitude from the vertex ofthe right angle of a right triangle to its hypotenuse is the geometricmean between the measures of the two segments of the hypotenuse.Is the converse of this theorem true? In order to find out, it will helpto rewrite the original theorem in if-then form as follows.

If !ABQ is a right triangle with right angle at Q, then QP is the geometric mean between AP and PB, where Pis between A and B and Q"P" is perpendicular to A"B".

1. Write the converse of the if-then form of the theorem.

2. Is the converse of the original theorem true? Refer to the figure at the right to explain your answer.

You may find it interesting to examine the other theorems inChapter 7 to see whether their converses are true or false. You willneed to restate the theorems carefully in order to write theirconverses.

Q

BPA

Q

BPA

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

Study Guide and InterventionSpecial Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3

Properties of 45°-45°-90° Triangles The sides of a 45°-45°-90° right triangle have aspecial relationship.

If the leg of a 45°-45°-90°right triangle is x units, show that the hypotenuse is x#2! units.

Using the Pythagorean Theorem with a " b " x, then

c2 " a2 % b2

" x2 % x2

" 2x2

c " !2x2"" x!2"

x!%

x

x 245#

45#

In a 45°-45°-90° right triangle the hypotenuse is #2! times the leg. If the hypotenuse is 6 units,find the length of each leg.The hypotenuse is !2" times the leg, sodivide the length of the hypotenuse by !2".

a "

"

"

" 3!2" units

6!2"!2

6!2"!!2"!2"

6!!2"


ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6.

7. Find the perimeter of a square with diagonal 12 centimeters.

8. Find the diagonal of a square with perimeter 20 inches.

9. Find the diagonal of a square with perimeter 28 meters.

x 3!%2x 18x x

18

x10x

45#3!%2

x

8

45#

45#


Properties of 30°-60°-90° Triangles The sides of a 30°-60°-90° right triangle alsohave a special relationship.

In a 30°-60°-90° right triangle, show that the hypotenuse is twice the shorter leg and the longer leg is #3! times the shorter leg.

!MNQ is a 30°-60°-90° right triangle, and the length of the hypotenuse M"N" is two times the length of the shorter side N"Q".Using the Pythagorean Theorem,a2 " (2x) 2 # x2

" 4x2 # x2

" 3x2

a " !3x2"" x!3"

In a 30°-60°-90° right triangle, the hypotenuse is 5 centimeters.Find the lengths of the other two sides of the triangle.If the hypotenuse of a 30°-60°-90° right triangle is 5 centimeters, then the length of theshorter leg is half of 5 or 2.5 centimeters. The length of the longer leg is !3" times the length of the shorter leg, or (2.5)(!3") centimeters.

Find x and y.

1. 2. 3.

4. 5. 6.

7. The perimeter of an equilateral triangle is 32 centimeters. Find the length of an altitudeof the triangle to the nearest tenth of a centimeter.

8. An altitude of an equilateral triangle is 8.3 meters. Find the perimeter of the triangle tothe nearest tenth of a meter.

xy

60#

20

x y60#

12

x y

30#

9!%3

x

y

11

30#

x

y

60#

8

x

y30#

60#12

x

a

N

Q

P

M

2x30#30#

60#

60#


Special Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

ExercisesExercises

Example 1Example 1

Example 2Example 2

!MNP is an equilateraltriangle.!MNQ is a 30°-60°-90°right triangle.

Skills PracticeSpecial Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3

Find x and y.

1. 2. 3.

4. 5. 6.

For Exercises 7–9, use the figure at the right.

7. If a " 11, find b and c.

8. If b " 15, find a and c.

9. If c " 9, find a and b.

For Exercises 10 and 11, use the figure at the right.

10. The perimeter of the square is 30 inches. Find the length of B"C".

11. Find the length of the diagonal B"D".

12. The perimeter of the equilateral triangle is 60 meters. Find the length of an altitude.

13. !GEC is a 30°-60°-90° triangle with right angle at E, and E"C" is the longer leg. Find the coordinates of G in Quadrant I for E(1, 1) and C(4, 1).

E

FGD 60#

A B

CD 45#

bA

B

C

ac 60#

30#

yx#13

1313

13

y

x60#

16

y

x

45# 8

y

x45#

12

y

x

30#

32

y

x60# 24


Find x and y.

1. 2. 3.

4. 5. 6.

For Exercises 7–9, use the figure at the right.

7. If a " 4!3", find b and c.

8. If x " 3!3", find a and CD.

9. If a " 4, find CD, b, and y.

10. The perimeter of an equilateral triangle is 39 centimeters. Find the length of an altitudeof the triangle.

11. !MIP is a 30°-60°-90° triangle with right angle at I, and I"P" the longer leg. Find thecoordinates of M in Quadrant I for I(3, 3) and P(12, 3).

12. !TJK is a 45°-45°-90° triangle with right angle at J. Find the coordinates of T inQuadrant II for J(#2, #3) and K(3, #3).

13. BOTANICAL GARDENS One of the displays at a botanical garden is an herb garden planted in the shape of a square. The square measures 6 yards on each side. Visitors can view the herbs from adiagonal pathway through the garden. How long is the pathway?

6 yd 6 yd

6 yd

6 yd

bA

B

C

Da

x

y60#

30#

c

x45#

11

y60#3.5

xy

x#y 28

y

x30#

26y

x2560#

yx

45#9

Practice Special Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

Reading to Learn MathematicsSpecial Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3

Pre-Activity How is triangle tiling used in wallpaper design?

Read the introduction to Lesson 7-3 at the top of page 357 in your textbook.• How can you most completely describe the larger triangle and the two

smaller triangles in tile 15?

• How can you most completely describe the larger triangle and the twosmaller triangles in tile 16? (Include angle measures in describing all thetriangles.)

Reading the Lesson1. Supply the correct number or numbers to complete each statement.

a. In a 45°-45°-90° triangle, to find the length of the hypotenuse, multiply the length of a

leg by .

b. In a 30°-60°-90° triangle, to find the length of the hypotenuse, multiply the length of

the shorter leg by .

c. In a 30°-60°-90° triangle, the longer leg is opposite the angle with a measure of .

d. In a 30°-60°-90° triangle, to find the length of the longer leg, multiply the length of

the shorter leg by .

e. In an isosceles right triangle, each leg is opposite an angle with a measure of .

f. In a 30°-60°-90° triangle, to find the length of the shorter leg, divide the length of the

longer leg by .

g. In 30°-60°-90° triangle, to find the length of the longer leg, divide the length of the

hypotenuse by and multiply the result by .

h. To find the length of a side of a square, divide the length of the diagonal by .

2. Indicate whether each statement is always, sometimes, or never true.a. The lengths of the three sides of an isosceles triangle satisfy the Pythagorean

Theorem.b. The lengths of the sides of a 30°-60°-90° triangle form a Pythagorean triple.c. The lengths of all three sides of a 30°-60°-90° triangle are positive integers.

Helping You Remember3. Some students find it easier to remember mathematical concepts in terms of specific

numbers rather than variables. How can you use specific numbers to help you rememberthe relationship between the lengths of the three sides in a 30°-60°-90° triangle?


Constructing Values of Square RootsThe diagram at the right shows a right isosceles triangle with two legs of length 1 inch. By the Pythagorean Theorem, the length of the hypotenuse is !2" inches. By constructing an adjacent right triangle with legs of !2" inches and 1 inch, you can create a segment of length !3".

By continuing this process as shown below, you can construct a “wheel” of square roots. This wheel is called the “Wheel of Theodorus”after a Greek philosopher who lived about 400 B.C.

Continue constructing the wheel until you make a segment oflength !18".

!%

1

1

1

3!%

2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

1

1

11

1

1

!%2

!%3

!%5

!%6

!%7

!%8

!"10

!"11 !"12!"13

!"14

!"15

!"17

!"18

!"16 " 4

!%4 " 2

!%9 " 3

Study Guide and InterventionTrigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan,respectively.

sin R " !leg

hyopppotoesnitues#

eR

! cos R " tan R "

" !rt! " !

st! " !

rs!

Find sin A, cos A, and tan A. Express each ratio as a decimal to the nearest thousandth.

sin A " !ohpyppoostietneulseeg

! cos A " !ahdyjpaocteenntulseeg

! tan A " !aopd

pja

ocseintet

lleegg!

" !BAB

C! " !A

ABC! " !

BAC

C!

" !153! " !

11

23! " !1

52!

# 0.385 # 0.923 # 0.417

Find the indicated trigonometric ratio as a fraction and as a decimal. If necessary, round to the nearest ten-thousandth.

1. sin A 2. tan B

3. cos A 4. cos B

5. sin D 6. tan E

7. cos E 8. cos D

16

1620

12

3430

C

B

A D F

E

12

135C

B

A

leg opposite #R!!!leg adjacent to #R

leg adjacent to #R!!!hypotenuse

s

tr

T

S

R

ExercisesExercises

ExampleExample


Use Trigonometric Ratios In a right triangle, if you know the measures of two sidesor if you know the measures of one side and an acute angle, then you can use trigonometricratios to find the measures of the missing sides or angles of the triangle.

Find x, y, and z. Round each measure to the nearest whole number. 1858#

x # CB y

zA


Trigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

a. Find x.

x % 58 " 90x " 32

b. Find y.

tan A " !1y8!

tan 58° " !1y8!

y " 18 tan 58°y # 29

c. Find z.

cos A " !1z8!

cos 58° " !1z8!

z cos 58° " 18

z " !cos18

58°!

z # 34

ExercisesExercises

Find x. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.15

64# x1640#

x

4

1x#12

5x#

12 16

x#3228#

x

ExampleExample

Skills PracticeTrigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Use !RST to find sin R, cos R, tan R, sin S, cos S, and tan S.Express each ratio as a fraction and as a decimal to the nearest hundredth.

1. r " 16, s " 30, t " 34 2. r " 10, s " 24, t " 26

Use a calculator to find each value. Round to the nearest ten-thousandth.

3. sin 5 4. tan 23 5. cos 61

6. sin 75.8 7. tan 17.3 8. cos 52.9

Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to thenearest ten-thousandth.

9. tan C 10. sin A 11. cos C

Find the measure of each acute angle to the nearest tenth of a degree.

12. sin B " 0.2985 13. tan A " 0.4168 14. cos R " 0.8443

15. tan C " 0.3894 16. cos B " 0.7329 17. sin A " 0.1176


18. 19. 20.

19

x

33# UL

S

27

x #8

BA

C27

x #

13

BA

C

41

409B

AC

sR

S

T

rt


Use !LMN to find sin L, cos L, tan L, sin M, cos M, and tan M.Express each ratio as a fraction and as a decimal to the nearest hundredth.

1. ! " 15, m " 36, n" 39 2. ! " 12, m " 12!3", n " 24

Use a calculator to find each value. Round to the nearest ten-thousandth.

3. sin 92.4 4. tan 27.5 5. cos 64.8

Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to the nearest ten-thousandth.

6. cos A 7. tan B 8. sin A

Find the measure of each acute angle to the nearest tenth of a degree.

9. sin B " 0.7823 10. tan A " 0.2356 11. cos R " 0.6401


12. 13. 14.

15. GEOGRAPHY Diego used a theodolite to map a region of land for his class in geomorphology. To determine the elevation of a vertical rockformation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters and the angle was 36 degrees. What is the height of the formation to the nearest meter?

36#43 m

41#x

3229

x #9

23

x #

11

15

5!"105

CA

B

ML

N

Practice Trigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

Reading to Learn MathematicsTrigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Pre-Activity How can surveyors determine angle measures?


• Why is it important to determine the relative positions accurately innavigation? (Give two possible reasons.)

• What does calibrated mean?

Reading the Lesson

1. Refer to the figure. Write a ratio using the side lengths in the figure to represent each of the following trigonometric ratios.

A. sin N B. cos N

C. tan N D. tan M

E. sin M F. cos M

2. Assume that you enter each of the expressions in the list on the left into your calculator.Match each of these expressions with a description from the list on the right to tell whatyou are finding when you enter this expression.

P

M N

a. sin 20

b. cos 20

c. sin#1 0.8

d. tan#1 0.8

e. tan 20

f. cos#1 0.8

i. the degree measure of an acute angle whose cosine is 0.8

ii. the ratio of the length of the leg adjacent to the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle

iii.the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe adjacent leg is 0.8

iv. the ratio of the length of the leg opposite the 20° angle to thelength of the leg adjacent to it in a 20°-70°-90° triangle

v. the ratio of the length of the leg opposite the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle

vi. the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe hypotenuse is 0.8


3. How can the co in cosine help you to remember the relationship between the sines andcosines of the two acute angles of a right triangle?


Sine and Cosine of AnglesThe following diagram can be used to obtain approximate values for the sineand cosine of angles from 0° to 90°. The radius of the circle is 1. So, the sineand cosine values can be read directly from the vertical and horizontal axes.

Find approximate values for sin 40°and cos 40#. Consider the triangle formed by the segment marked 40°, as illustrated by the shaded triangle at right.

sin 40° " !ac! # !

0.164! or 0.64 cos 40° " !

bc! # !

0.177! or 0.77

1. Use the diagram above to complete the chart of values.

2. Compare the sine and cosine of two complementary angles (angles whose sum is 90°). What do you notice?

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

90°

0°

10°

20°

30°

40°

50°

60°

70°80°

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

x° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90°

sin x° 0.64

cos x° 0.77

1

0

40°0.64

c " 1 unit

x °b " cos x ° 0.77 1

a " sin x °

ExampleExample

Study Guide and InterventionAngles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Angles of Elevation Many real-world problems that involve looking up to an object can be described in terms of an angle of elevation, which is the angle between an observer’s line of sight and a horizontal line.

The angle of elevation from point A to the top of a cliff is 34°. If point A is 1000 feet from the base of the cliff,how high is the cliff?Let x " the height of the cliff.

tan 34° " !10x00! tan " !

oapdpjaocseitnet!

1000(tan 34°) " x Multiply each side by 1000.

674.5 " x Use a calculator.

The height of the cliff is about 674.5 feet.

Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.

1. The angle of elevation from point A to the top of a hill is 49°.If point A is 400 feet from the base of the hill, how high is the hill?

2. Find the angle of elevation of the sun when a 12.5-meter-tall telephone pole casts a 18-meter-long shadow.

3. A ladder leaning against a building makes an angle of 78°with the ground. The foot of the ladder is 5 feet from the building. How long is the ladder?

4. A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower.That person observes an air traffic controller at the window of the 132-foot tower. What is the angle of elevation?

?5 ft

100 ft

132 ft

78#5 ft

?

18 m

12.5 m

sun

?

✹

400 ft

?

49#A

?

1000 ft34#A

x

angle ofelevation

line of si

ght

ExercisesExercises

ExampleExample


Angles of Depression When an observer is looking down, the angle of depression is the angle between the observer’s line of sight and a horizontal line.

The angle of depression from the top of an 80-foot building to point A on the ground is 42°. How far is the foot of the building from point A?Let x " the distance from point A to the foot of the building. Since the horizontal line is parallel to the ground, the angle of depression#DBA is congruent to #BAC.

tan 42° " !8x0! tan " !

oa

pd

pja

ocseitnet!

x(tan 42°) " 80 Multiply each side by x.

x " !tan80

42°! Divide each side by tan 42°.

x # 88.8 Use a calculator.

Point A is about 89 feet from the base of the building.

Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.

1. The angle of depression from the top of a sheer cliff to point A on the ground is 35°. If point A is 280 feet from the base of the cliff, how tall is the cliff?

2. The angle of depression from a balloon on a 75-foot string to a person on the ground is 36°. How high is the balloon?

3. A ski run is 1000 yards long with a vertical drop of 208 yards. Find the angle of depression from the top of the ski run to the bottom.

4. From the top of a 120-foot-high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 19°. How far from the base of thetower is the airplane?

120 ft

?

19#

208 yd

?

1000 yd

36#

75 ft ?

A

35#

280 ft

?

A C

BD

x42#

angle ofdepression

horizontal

80 ft

Yline of sight

horizontalangle ofdepression


Angles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

ExercisesExercises

ExampleExample

Skills PracticeAngles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Name the angle of depression or angle of elevation in each figure.

1. 2.

3. 4.

5. MOUNTAIN BIKING On a mountain bike trip along the Gemini Bridges Trail in Moab,Utah, Nabuko stopped on the canyon floor to get a good view of the twin sandstonebridges. Nabuko is standing about 60 meters from the base of the canyon cliff, and thenatural arch bridges are about 100 meters up the canyon wall. If her line of sight is fivefeet above the ground, what is the angle of elevation to the top of the bridges? Round tothe nearest tenth degree.

6. SHADOWS Suppose the sun casts a shadow off a 35-foot building.If the angle of elevation to the sun is 60°, how long is the shadow to the nearest tenth of a foot?

7. BALLOONING From her position in a hot-air balloon, Angie can see her car parked in afield. If the angle of depression is 8° and Angie is 38 meters above the ground, what isthe straight-line distance from Angie to her car? Round to the nearest whole meter.

8. INDIRECT MEASUREMENT Kyle is at the end of a pier 30 feet above the ocean. His eye level is 3 feet above the pier. He is using binoculars to watch a whale surface. If the angle of depression of the whale is 20°, how far is the whale from Kyle’s binoculars? Round to the nearest tenth foot.

whale water level

20#Kyle’s eyes

pier3 ft

30 ft

60#?

35 ft

Z

P

W

R

D

A

C

B

T

W

R

S

F

T

L

S


Name the angle of depression or angle of elevation in each figure.

1. 2.

3. WATER TOWERS A student can see a water tower from the closest point of the soccerfield at San Lobos High School. The edge of the soccer field is about 110 feet from thewater tower and the water tower stands at a height of 32.5 feet. What is the angle ofelevation if the eye level of the student viewing the tower from the edge of the soccerfield is 6 feet above the ground? Round to the nearest tenth degree.

4. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladderreaches a 30-foot roof that needs repair. If the angle of elevation from the bottom of theladder to the roof is 55°, how far is the ladder from the base of the wall? Round youranswer to the nearest foot.

5. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet on any property. Lindsay wants to determinewhether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from theflagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth foot?

6. GEOGRAPHY Stephan is standing on a mesa at the Painted Desert. The elevation ofthe mesa is about 1380 meters and Stephan’s eye level is 1.8 meters above ground. IfStephan can see a band of multicolored shale at the bottom and the angle of depressionis 29°, about how far is the band of shale from his eyes? Round to the nearest meter.

7. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40-foot ocean bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his dogs are 34°and 48°, how far apart are the dogs to the nearest foot?

48# 34#

40 ft

6 ft

Mr. Dominguez

bluff

25#5.5 ft

36 ft

x

R

M

P

L

T

Y

R

Z

Practice Angles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

Reading to Learn MathematicsAngles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Pre-Activity How do airline pilots use angles of elevation and depression?


What does the angle measure tell the pilot?

Reading the Lesson

1. Refer to the figure. The two observers are looking at one another. Select the correct choice for each question.

a. What is the line of sight?(i) line RS (ii) line ST (iii) line RT (iv) line TU

b. What is the angle of elevation?(i) #RST (ii) #SRT (iii) #RTS (iv) #UTR

c. What is the angle of depression?(i) #RST (ii) #SRT (iii) #RTS (iv) #UTR

d. How are the angle of elevation and the angle of depression related?(i) They are complementary.(ii) They are congruent.(iii) They are supplementary.(iv) The angle of elevation is larger than the angle of depression.

e. Which postulate or theorem that you learned in Chapter 3 supports your answer forpart c?(i) Corresponding Angles Postulate(ii) Alternate Exterior Angles Theorem(iii) Consecutive Interior Angles Theorem(iv) Alternate Interior Angles Theorem

2. A student says that the angle of elevation from his eye to the top of a flagpole is 135°.What is wrong with the student’s statement?


3. A good way to remember something is to explain it to someone else. Suppose a classmatefinds it difficult to distinguish between angles of elevation and angles of depression. Whatare some hints you can give her to help her get it right every time?

S

T observer attop of building

observeron ground R

U


Reading MathematicsThe three most common trigonometric ratios are sine, cosine, and tangent. Three other ratios are thecosecant, secant, and cotangent. The chart below shows abbreviations and definitions for all six ratios.Refer to the triangle at the right.

Use the abbreviations to rewrite each statement as an equation.

1. The secant of angle A is equal to 1 divided by the cosine of angle A.

2. The cosecant of angle A is equal to 1 divided by the sine of angle A.

3. The cotangent of angle A is equal to 1 divided by the tangent of angle A.

4. The cosecant of angle A multiplied by the sine of angle A is equal to 1.

5. The secant of angle A multiplied by the cosine of angle A is equal to 1.

6. The cotangent of angle A times the tangent of angle A is equal to 1.

Use the triangle at right. Write each ratio.

7. sec R 8. csc R 9. cot R

10. sec S 11. csc S 12. cot S

13. If sin x° " 0.289, find the value of csc x°.

14. If tan x° " 1.376, find the value of cot x°.

R

T S

ts

r

A

ca

b

B

C

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

Abbreviation Read as: Ratio

sin A the sine of #A " !ac!

cos A the cosine of #A " !bc!

tan A the tangent of #A " !ab!

csc A the cosecant of #A " !ac

!

sec A the secant of #A " !bc

!

cot A the cotangent of #A " !ba!

leg adjacent to #A!!!

leg opposite #A

hypotenuse!!!leg adjacent to #A

hypotenuse!!leg opposite #A

leg opposite #A!!!leg adjacent to #A

leg adjacent to #A!!!hypotenuse

leg opposite #A!!hypotenuse

Study Guide and InterventionThe Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6


Less

on

7-6

The Law of Sines In any triangle, there is a special relationship between the angles ofthe triangle and the lengths of the sides opposite the angles.

Law of Sines !sin

aA

! " !sin

bB

! " !sin

cC

!

In !ABC, find b.

!sin

cC

! " !sin

bB

! Law of Sines

!sin

3045°! " !

sinb74°! m#C " 45, c " 30, m#B " 74

b sin 45° " 30 sin 74° Cross multiply.

b " !30

sisnin45

7°4°

! Divide each side by sin 45°.

b # 40.8 Use a calculator.

45#

3074#

b

B

AC

In !DEF, find m"D.

!sin

dD

! " !sin

eE

! Law of Sines

!si

2n8D

! " !sin

2458°!

d " 28, m#E " 58, e " 24

24 sin D " 28 sin 58° Cross multiply.

sin D " !28 s

2in4

58°! Divide each side by 24.

D " sin#1 !28 s

2in4

58°! Use the inverse sine.

D # 81.6° Use a calculator.

58#

24

28

E

FD


ExercisesExercises

Find each measure using the given measures of !ABC. Round angle measures tothe nearest degree and side measures to the nearest tenth.

1. If c " 12, m#A " 80, and m#C " 40, find a.

2. If b " 20, c " 26, and m#C " 52, find m#B.

3. If a " 18, c " 16, and m#A " 84, find m#C.

4. If a " 25, m#A " 72, and m#B " 17, find b.

5. If b " 12, m#A " 89, and m#B" 80, find a.

6. If a " 30, c " 20, and m#A " 60, find m#C.


Use the Law of Sines to Solve Problems You can use the Law of Sines to solvesome problems that involve triangles.

Law of SinesLet !ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measures A, B, and C, respectively. Then !sin

aA! " !sin

bB! " !sin

cC!.

Isosceles !ABC has a base of 24 centimeters and a vertex angle of 68°. Find the perimeter of the triangle.The vertex angle is 68°, so the sum of the measures of the base angles is 112 and m#A " m#C " 56.

!sin

bB

! " !sin

aA

! Law of Sines

!sin

2468°! " !

sina56°! m#B " 68, b " 24, m#A " 56

a sin 68° " 24 sin 56° Cross multiply.

a " !24

sisnin68

5°6°

! Divide each side by sin 68°.

# 21.5 Use a calculator.

The triangle is isosceles, so c " 21.5.The perimeter is 24 % 21.5 % 21.5 or about 67 centimeters.

Draw a triangle to go with each exercise and mark it with the given information.Then solve the problem. Round angle measures to the nearest degree and sidemeasures to the nearest tenth.

1. One side of a triangular garden is 42.0 feet. The angles on each end of this side measure66° and 82°. Find the length of fence needed to enclose the garden.

2. Two radar stations A and B are 32 miles apart. They locate an airplane X at the sametime. The three points form #XAB, which measures 46°, and #XBA, which measures52°. How far is the airplane from each station?

3. A civil engineer wants to determine the distances from points A and B to an inaccessiblepoint C in a river. #BAC measures 67° and #ABC measures 52°. If points A and B are82.0 feet apart, find the distance from C to each point.

4. A ranger tower at point A is 42 kilometers north of a ranger tower at point B. A fire atpoint C is observed from both towers. If #BAC measures 43° and #ABC measures 68°,which ranger tower is closer to the fire? How much closer?

68#

bc a

24

B

CA


The Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6

ExampleExample

ExercisesExercises

Skills PracticeThe Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6


Less

on

7-6

Find each measure using the given measures from !ABC. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.

1. If m#A " 35, m#B " 48, and b " 28, find a.

2. If m#B " 17, m#C " 46, and c " 18, find b.

3. If m#C " 86, m#A " 51, and a " 38, find c.

4. If a " 17, b " 8, and m#A " 73, find m#B.

5. If c " 38, b " 34, and m#B " 36, find m#C.

6. If a " 12, c " 20, and m#C " 83, find m#A.

7. If m#A " 22, a " 18, and m#B" 104, find b.

Solve each !PQR described below. Round measures to the nearest tenth.

8. p " 27, q " 40, m#P " 33

9. q " 12, r " 11, m#R " 16

10. p " 29, q " 34, m#Q " 111

11. If m#P " 89, p " 16, r " 12

12. If m#Q " 103, m#P " 63, p " 13

13. If m#P " 96, m#R " 82, r " 35

14. If m#R " 49, m#Q " 76, r " 26

15. If m#Q " 31, m#P " 52, p " 20

16. If q " 8, m#Q " 28, m#R " 72

17. If r " 15, p " 21, m#P " 128


Find each measure using the given measures from !EFG. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.

1. If m#G " 14, m#E " 67, and e " 14, find g.

2. If e " 12.7, m#E " 42, and m#F " 61, find f.

3. If g " 14, f " 5.8, and m#G " 83, find m#F.

4. If e " 19.1, m#G " 34, and m#E " 56, find g.

5. If f " 9.6, g " 27.4, and m#G " 43, find m#F.

Solve each !STU described below. Round measures to the nearest tenth.

6. m#T " 85, s " 4.3, t " 8.2

7. s " 40, u " 12, m#S " 37

8. m#U " 37, t " 2.3, m#T " 17

9. m#S " 62, m#U " 59, s " 17.8

10. t " 28.4, u " 21.7, m#T " 66

11. m#S " 89, s " 15.3, t " 14

12. m#T " 98, m#U " 74, u " 9.6

13. t " 11.8, m#S " 84, m#T " 47

14. INDIRECT MEASUREMENT To find the distance from the edge of the lake to the tree on the island in the lake, Hannah set up atriangular configuration as shown in the diagram. The distance from location A to location B is 85 meters. The measures of the angles at A and B are 51° and 83°, respectively. What is the distancefrom the edge of the lake at B to the tree on the island at C?

A

C

B

Practice The Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6

Reading to Learn MathematicsThe Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6


Less

on

7-6

Pre-Activity How are triangles used in radio astronomy?


Why might several antennas be better than one single antenna whenstudying distant objects?

Reading the Lesson

1. Refer to the figure. According to the Law of Sines, which of the following are correct statements?

A. !sinm

M! " !sinn

N! " !sinp

P! B. !sin

Mm

! " !si

Nn n! " !

sinP

p!

C. !co

ms M! " !

cosn

N! " !

cops P! D. !

sinm

M! % !

sinn

N! " !

sinp

P!

E. (sin M)2 % (sin N)2 " (sin P)2 F. !sin

pP

! " !sin

mM

! " !sin

nN

!

2. State whether each of the following statements is true or false. If the statement is false,explain why.

a. The Law of Sines applies to all triangles.

b. The Pythagorean Theorem applies to all triangles.

c. If you are given the length of one side of a triangle and the measures of any twoangles, you can use the Law of Sines to find the lengths of the other two sides.

d. If you know the measures of two angles of a triangle, you should use the Law of Sinesto find the measure of the third angle.

e. A friend tells you that in triangle RST, m#R " 132, r " 24 centimeters, and s " 31centimeters. Can you use the Law of Sines to solve the triangle? Explain.


3. Many students remember mathematical equations and formulas better if they can statethem in words. State the Law of Sines in your own words without using variables ormathematical symbols.

P

M Np

mn


IdentitiesAn identity is an equation that is true for all values of the variable for which both sides are defined. One way to verify an identity is to use a right triangle and the definitions fortrigonometric functions.

Verify that (sin A)2 $ (cos A)2 " 1 is an identity.

(sin A)2 % (cos A)2 " &!ac!'2 % &!

bc!'2

" !a2 %

cb2

! " !cc

2

2! " 1

To check whether an equation may be an identity, you can testseveral values. However, since you cannot test all values, youcannot be certain that the equation is an identity.

Test sin 2x " 2 sin x cos x to see if it could be an identity.

Try x " 20. Use a calculator to evaluate each expression.

sin 2x " sin 40 2 sin x cos x " 2 (sin 20)(cos 20)# 0.643 # 2(0.342)(0.940)

# 0.643

Since the left and right sides seem equal, the equation may be an identity.

Use triangle ABC shown above. Verify that each equation is an identity.

1. !csoins

AA

! " !tan1

A! 2. !tsainn

BB

! " !co1s B!

3. tan B cos B " sin B 4. 1 # (cos B)2 " (sin B)2

Try several values for x to test whether each equation could be an identity.

5. cos 2x " (cos x)2 # (sin x)2 6. cos (90 # x) " sin x

B

A C

ca

b

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6

Example 1Example 1

Example 2Example 2

Study Guide and InterventionThe Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7


Less

on

7-7

The Law of Cosines Another relationship between the sides and angles of any triangleis called the Law of Cosines. You can use the Law of Cosines if you know three sides of atriangle or if you know two sides and the included angle of a triangle.

Let !ABC be any triangle with a, b, and c representing the measures of the sides opposite Law of Cosines the angles with measures A, B, and C, respectively. Then the following equations are true.

a2 " b2 % c2 # 2bc cos A b2 " a2 % c2 # 2ac cos B c2 " a2 % b2 # 2ab cos C

In !ABC, find c.c2 " a2 % b2 # 2ab cos C Law of Cosines

c2 " 122 % 102 # 2(12)(10)cos 48° a " 12, b " 10, m#C " 48

c " !122 %" 102 #" 2(12)"(10)co"s 48°" Take the square root of each side.

c # 9.1 Use a calculator.

In !ABC, find m"A.a2 " b2 % c2 # 2bc cos A Law of Cosines

72 " 52 % 82 # 2(5)(8) cos A a " 7, b " 5, c " 8

49 " 25 % 64 # 80 cos A Multiply.

#40 " #80 cos A Subtract 89 from each side.

!12! " cos A Divide each side by #80.

cos#1 !12! " A Use the inverse cosine.

60° " A Use a calculator.

Find each measure using the given measures from !ABC. Round angle measuresto the nearest degree and side measures to the nearest tenth.

1. If b " 14, c " 12, and m#A " 62, find a.

2. If a " 11, b " 10, and c " 12, find m#B.

3. If a " 24, b " 18, and c " 16, find m#C.

4. If a " 20, c " 25, and m#B " 82, find b.

5. If b " 18, c " 28, and m#A " 59, find a.

6. If a " 15, b " 19, and c " 15, find m#C.

58

7 CB

A

48#12 10

c

C

BA

Example 1Example 1

Example 2Example 2

ExercisesExercises


Use the Law of Cosines to Solve Problems You can use the Law of Cosines tosolve some problems involving triangles.

Let !ABC be any triangle with a, b, and c representing the measures of the sides opposite the Law of Cosines angles with measures A, B, and C, respectively. Then the following equations are true.

a2 " b2 % c2 # 2bc cos A b2 " a2 % c2 # 2ac cos B c2 " a2 % b2 # 2ab cos C

Ms. Jones wants to purchase a piece of land with the shape shown. Find the perimeter of the property.Use the Law of Cosines to find the value of a.

a2 " b2 % c2 # 2bc cos A Law of Cosines

a2 " 3002 % 2002 # 2(300)(200) cos 88° b " 300, c " 200, m#A " 88

a " !130,0"00 #"120,0"00 cos" 88°" Take the square root of each side.


Use the Law of Cosines again to find the value of c.

c2 " a2 % b2 # 2ab cos C Law of Cosines

c2 " 354.72 % 3002 # 2(354.7)(300) cos 80° a " 354.7, b " 300, m#C " 80

c " !215,8"12.09" # 21"2,820" cos 8"0°" Take the square root of each side.


The perimeter of the land is 300 % 200 % 422.9 % 200 or about 1223 feet.

Draw a figure or diagram to go with each exercise and mark it with the giveninformation. Then solve the problem. Round angle measures to the nearest degreeand side measures to the nearest tenth.

1. A triangular garden has dimensions 54 feet, 48 feet, and 62 feet. Find the angles at eachcorner of the garden.

2. A parallelogram has a 68° angle and sides 8 and 12. Find the lengths of the diagonals.

3. An airplane is sighted from two locations, and its position forms an acute triangle withthem. The distance to the airplane is 20 miles from one location with an angle ofelevation 48.0°, and 40 miles from the other location with an angle of elevation of 21.8°.How far apart are the two locations?

4. A ranger tower at point A is directly north of a ranger tower at point B. A fire at point Cis observed from both towers. The distance from the fire to tower A is 60 miles, and thedistance from the fire to tower B is 50 miles. If m#ACB " 62, find the distance betweenthe towers.

200 ft

300 ft

300 ft

88#

80#ca


The Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7

ExampleExample

ExercisesExercises

Skills PracticeThe Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7


Less

on

7-7

In !RST, given the following measures, find the measure of the missing side.

1. r " 5, s " 8, m#T " 39

2. r " 6, t " 11, m#S " 87

3. r " 9, t " 15, m#S " 103

4. s " 12, t " 10, m#R " 58

In !HIJ, given the lengths of the sides, find the measure of the stated angle to thenearest tenth.

5. h " 12, i " 18, j " 7; m#H

6. h " 15, i " 16, j " 22; m#I

7. h " 23, i " 27, j " 29; m#J

8. h " 37, i " 21, j " 30; m#H

Determine whether the Law of Sines or the Law of Cosines should be used first tosolve each triangle. Then solve each triangle. Round angle measures to the nearestdegree and side measures to the nearest tenth.

9. 10.

11. a " 10, b " 14, c "19 12. a " 12, b " 10, m#C " 27

Solve each !RST described below. Round measures to the nearest tenth.

13. r " 12, s " 32, t " 34

14. r " 30, s " 25, m#T " 42

15. r " 15, s " 11, m#R " 67

16. r " 21, s " 28, t " 30

M

L N

!86#

52

24

B

A C

c

66#

33

19


In !JKL, given the following measures, find the measure of the missing side.

1. j " 1.3, k " 10, m#L " 77

2. j " 9.6, ! " 1.7, m#K " 43

3. j " 11, k " 7, m#L " 63

4. k " 4.7, ! " 5.2, m#J " 112

In !MNQ, given the lengths of the sides, find the measure of the stated angle tothe nearest tenth.

5. m " 17, n " 23, q " 25; m#Q

6. m " 24, n " 28, q " 34; m#M

7. m " 12.9, n " 18, q " 20.5; m#N

8. m " 23, n " 30.1, q " 42; m#Q

Determine whether the Law of Sines or the Law of Cosines should be used first tosolve !ABC. Then sole each triangle. Round angle measures to the nearest degreeand side measure to the nearest tenth.

9. a " 13, b " 18, c " 19 10. a " 6, b " 19, m#C " 38

11. a " 17, b " 22, m#B " 49 12. a " 15.5, b " 18, m#C " 72

Solve each !FGH described below. Round measures to the nearest tenth.

13. m#F " 54, f " 12.5, g " 11

14. f "20, g " 23, m#H " 47

15. f " 15.8, g " 11, h " 14

16. f " 36, h " 30, m#G " 54

17. REAL ESTATE The Esposito family purchased a triangular plot of land on which theyplan to build a barn and corral. The lengths of the sides of the plot are 320 feet, 286 feet,and 305 feet. What are the measures of the angles formed on each side of the property?

Practice The Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7

Reading to Learn MathematicsThe Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7


Less

on

7-7

Pre-Activity How are triangles used in building design?


What could be a disadvantage of a triangular room?

Reading the Lesson1. Refer to the figure. According to the Law of Cosines, which

statements are correct for !DEF?

A. d2 " e2 % f 2 # ef cos D B. e2 " d2 % f 2 # 2df cos E

C. d2 " e2 % f 2 % 2ef cos D D. f 2 " d2 % e2 # 2ef cos F

E. f2 " d2 % e2 # 2de cos F F. d2 " e2 % f 2

G. !sin

dD

! " !sin

eE

! " !sin

fF

! H. d " !e2 % f"2 # 2e"f cos "D"

2. Each of the following describes three given parts of a triangle. In each case, indicatewhether you would use the Law of Sines or the Law of Cosines first in solving a trianglewith those given parts. (In some cases, only one of the two laws would be used in solvingthe triangle.)

a. SSS b. ASA

c. AAS d. SAS

e. SSA

3. Indicate whether each statement is true or false. If the statement is false, explain why.

a. The Law of Cosines applies to right triangles.

b. The Pythagorean Theorem applies to acute triangles.

c. The Law of Cosines is used to find the third side of a triangle when you are given themeasures of two sides and the nonincluded angle.

d. The Law of Cosines can be used to solve a triangle in which the measures of the threesides are 5 centimeters, 8 centimeters, and 15 centimeters.

Helping You Remember4. A good way to remember a new mathematical formula is to relate it to one you already

know. The Law of Cosines looks somewhat like the Pythagorean Theorem. Both formulasmust be true for a right triangle. How can that be?

D

dE

e

F

f


Spherical TrianglesSpherical trigonometry is an extension of plane trigonometry.Figures are drawn on the surface of a sphere. Arcs of great circles correspond to line segments in the plane. The arcs of three great circles intersecting on a sphere form a spherical triangle. Angles have the same measure as the tangent lines drawn to each great circle at the vertex. Since the sides are arcs, they too can be measured in degrees.

Solve the spherical triangle given a " 72#,b " 105#, and c " 61#.

Use the Law of Cosines.

0.3090 " (–0.2588)(0.4848) % (0.9659)(0.8746) cos Acos A " 0.5143

A " 59°

#0.2588 " (0.3090)(0.4848) % (0.9511)(0.8746) cos Bcos B " #0.4912

B " 119°

0.4848 " (0.3090)(–0.2588) % (0.9511)(0.9659) cos Ccos C " 0.6148

C " 52°

Check by using the Law of Sines.

!ssiinn

75

29

°°! " !

ssiinn

11

01

59

°°! " !

ssiinn

65

12

°°! " 1.1

Solve each spherical triangle.

1. a " 56°, b " 53°, c " 94° 2. a " 110°, b " 33°, c " 97°

3. a " 76°, b " 110°, C " 49° 4. b " 94°, c " 55°, A " 48°

A

C

B

c

ba

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7

The sum of the sides of a spherical triangle is less than 360°.The sum of the angles is greater than 180° and less than 540°.The Law of Sines for spherical triangles is as follows.

!ssiinn

Aa

! " !ssiinn

Bb

! " !ssiinn

Cc

!

There is also a Law of Cosines for spherical triangles.cos a " cos b cos c % sin b sin c cos Acos b " cos a cos c % sin a sin c cos Bcos c " cos a cos b % sin a sin b cos C

ExampleExample

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.

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13

Answers (Lesson 7-1)


An

swer

s

Skil

ls P

ract

ice

Geo

met

ric M

ean

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-1

7-1

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Lesson 7-1

Fin

d t

he

geom

etri

c m

ean

bet

wee

n e

ach

pai

r of

nu

mbe

rs.S

tate

exa

ct a

nsw

ers

and

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ers

to t

he

nea

rest

ten

th.

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Fin

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17.3

10.C

IVIL

EN

GIN

EER

ING

An

airp

ort,

a fa

ctor

y,an

d a

shop

ping

cen

ter

are

at t

he v

erti

ces

of a

righ

t tr

iang

le fo

rmed

by

thre

e hi

ghw

ays.

The

air

port

and

fact

ory

are

6.0

mile

s ap

art.

The

irdi

stan

ces

from

the

sho

ppin

g ce

nter

are

3.6

mile

s an

d 4.

8 m

iles,

resp

ecti

vely

.A s

ervi

ce r

oad

will

be

cons

truc

ted

from

the

sho

ppin

g ce

nter

to

the

high

way

tha

t co

nnec

ts t

he a

irpo

rt a

ndfa

ctor

y.W

hat

is t

he s

hort

est

poss

ible

leng

th f

or t

he s

ervi

ce r

oad?

Rou

nd t

o th

e ne

ares

thu

ndre

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2.88

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z

20x

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3

z

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23

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8

17

6

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M

125

U

TA

V

Pra

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e (A

vera

ge)

Geo

met

ric M

ean

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-1

7-1



Rea

din

g t

o L

earn

Math

emati

csG

eom

etric

Mea

n

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-1

7-1

©G

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5G

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Lesson 7-1

Pre-

Act

ivit

yH

ow c

an t

he

geom

etri

c m

ean

be

use

d t

o vi

ew p

ain

tin

gs?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

1 at

the

top

of

page

342

in y

our

text

book

.

•W

hat

is a

dis

adva

ntag

e of

sta

ndin

g to

o cl

ose

to a

pai

ntin

g?Sa

mpl

e an

swer

:You

don

’t ge

t a g

ood

over

all v

iew.

•W

hat

is a

dis

adva

ntag

e of

sta

ndin

g to

o fa

r fr

om a

pai

ntin

g?Sa

mpl

e an

swer

:You

can

’t se

e al

l the

det

ails

in th

e pa

intin

g.R

ead

ing

th

e Le

sso

n1.

In t

he p

ast,

whe

n yo

u ha

ve s

een

the

wor

d m

ean

in m

athe

mat

ics,

it r

efer

red

to t

heav

erag

eor

ari

thm

etic

mea

nof

the

tw

o nu

mbe

rs.

a.C

ompl

ete

the

follo

win

g by

wri

ting

an

alge

brai

c ex

pres

sion

in e

ach

blan

k.

If a

and

bar

e tw

o po

siti

ve n

umbe

rs,t

hen

the

geom

etri

c m

ean

betw

een

aan

d b

is

and

thei

r ar

ithm

etic

mea

n is

.

b.E

xpla

in in

wor

ds,w

itho

ut u

sing

any

mat

hem

atic

al s

ymbo

ls,t

he d

iffe

renc

e be

twee

nth

e ge

omet

ric

mea

n an

d th

e al

gebr

aic

mea

n.Sa

mpl

e an

swer

:The

geo

met

ricm

ean

betw

een

two

num

bers

is th

e sq

uare

root

of t

heir

prod

uct.

The

arith

met

ic m

ean

of tw

o nu

mbe

rs is

hal

f the

ir su

m.

2.L

et r

and

sbe

tw

o po

siti

ve n

umbe

rs.I

n w

hich

of

the

follo

win

g eq

uati

ons

is z

equa

l to

the

geom

etri

c m

ean

betw

een

ran

d s?

A,C,

D,F

A.

! zs !"

!z r!B

.! zr !

"! zs !

C.

s:z

"z:

rD

.! zr !

"!z s!

E.

!z r!"

!z s!F.

!z s!"

! zr !

3.Su

pply

the

mis

sing

wor

ds o

r ph

rase

s to

com

plet

e th

e st

atem

ent

of e

ach

theo

rem

.

a.T

he m

easu

re o

f the

alt

itud

e dr

awn

from

the

ver

tex

of t

he r

ight

ang

le o

f a r

ight

tri

angl

e

to it

s hy

pote

nuse

is t

he

betw

een

the

mea

sure

s of

the

tw

o

segm

ents

of

the

.

b.If

the

alt

itud

e is

dra

wn

from

the

ver

tex

of t

he

angl

e of

a r

ight

tria

ngle

to

its

hypo

tenu

se,t

hen

the

mea

sure

of

a of

the

tri

angl

e

is t

he

betw

een

the

mea

sure

of t

he h

ypot

enus

e an

d th

e se

gmen

t

of t

he

adja

cent

to

that

leg.

c.If

the

alt

itud

e is

dra

wn

from

the

of

the

rig

ht a

ngle

of

a ri

ght

tria

ngle

to

its

,the

n th

e tw

o tr

iang

les

form

ed a

re

to t

he g

iven

tri

angl

e an

d to

eac

h ot

her.

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

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rem

embe

r a

new

mat

hem

atic

al c

once

pt is

to

rela

te it

to

som

ethi

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oual

read

y kn

ow.H

ow c

an t

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ng o

f mea

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a p

ropo

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n he

lp y

ou t

o re

mem

ber

how

to f

ind

the

geom

etri

c m

ean

betw

een

two

num

bers

?Sa

mpl

e an

swer

:Writ

e a

prop

ortio

n in

whi

ch th

e tw

o m

eans

are

equ

al.T

his

com

mon

mea

n is

the

geom

etric

mea

n be

twee

n th

e tw

o ex

trem

es.

sim

ilar

hypo

tenu

seve

rtex

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ric m

ean

leg

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omet

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Mat

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d M

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Pyt

hago

ras,

a G

reek

phi

loso

pher

who

live

d du

ring

the

six

th c

entu

ry B

.C.,

belie

ved

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re,b

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y,an

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who

le-

num

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Pyt

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out

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t tr

iang

les.

(The

sum

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the

squa

res

of t

he le

gs e

qual

s th

e sq

uare

of

the

hypo

tenu

se.)

But

Pyt

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ras

also

dis

cove

red

rela

tion

ship

s be

twee

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em

usic

al n

otes

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ale.

The

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hips

can

be

expr

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d as

rat

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men

t,T

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you

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lf s

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nces

you

fou

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Exe

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dete

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rich

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t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-1

7-1

3 4of

C s

tring



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

The

Pyth

agor

ean

Theo

rem

and

Its

Conv

erse

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-2

7-2

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Lesson 7-2

The

Pyth

ago

rean

Th

eore

mIn

a r

ight

tri

angl

e,th

e su

m o

f th

e

squa

res

of t

he m

easu

res

of t

he le

gs e

qual

s th

e sq

uare

of

the

mea

sure

of

the

hypo

tenu

se.

!A

BC

is a

rig

ht t

rian

gle,

so a

2%

b2"

c2.

Pro

ve t

he

Pyt

hag

orea

n T

heo

rem

.W

ith

alti

tude

C"D"

,eac

h le

g a

and

bis

a g

eom

etri

c m

ean

betw

een

hypo

tenu

se c

and

the

segm

ent

of t

he h

ypot

enus

e ad

jace

nt t

o th

at le

g.

! ac !"

!a y!an

d ! bc !

"!b x! ,

so a

2"

cyan

d b2

"cx

.

Add

the

tw

o eq

uati

ons

and

subs

titu

te c

"y

%x

to g

eta2

%b2

"cy

%cx

"c(

y%

x) "

c2.

cy

xa

b

hA

CBD

ca

bA

CB

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

a.F

ind

a.

a2%

b2"

c2Py

thag

orea

n Th

eore

m

a2%

122

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, c"

13

a2%

144

"16

9Si

mpl

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btra

ct.

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e sq

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side.

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1300

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Add.

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the

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re ro

ot o

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h sid

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36.1

#c

Use

a ca

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30

20

ACB

Exer

cises

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Fin

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.

1.2.

3.

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212

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,the

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, the

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er !

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c"

20

100

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0 "

400

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plify

.

400

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0✓Ad

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The

sum

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the

squa

res

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he t

wo

shor

ter

side

s eq

uals

the

squ

are

of t

he lo

nges

t si

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o th

etr

iang

le is

a r

ight

tri

angl

e.

Det

erm

ine

wh

eth

er e

ach

set

of

mea

sure

s ca

n b

e th

e m

easu

res

of t

he

sid

es o

f a

righ

t tr

ian

gle.

Th

en s

tate

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7-2

7-2



An

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Math

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Lesson 7-2

Pre-

Act

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ow a

re r

igh

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o bu

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spen

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e in

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n 7-

2 at

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top

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in y

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Do

the

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dra

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es a

re n

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opor

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l.In

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nger

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e la

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e lo

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is le

ss th

an tw

ice

the

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th o

f the

sho

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leg.

Rea

din

g t

he

Less

on

1.E

xpla

in in

you

r ow

n w

ords

the

dif

fere

nce

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een

how

the

Pyt

hago

rean

The

orem

is u

sed

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Pyt

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orem

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ther

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le w

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ght t

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2.R

efer

to

the

figu

re.F

or t

his

figu

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sta

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tru

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Pyt

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ngth

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d th

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AP

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PB

,whe

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d B

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the

theo

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.

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mea

n be

twee

n AP

and

PB,w

here

Pis

bet

wee

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and

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P "#

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t tria

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.

2.Is

the

con

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ori

gina

l the

orem

tru

e? R

efer

to

the

fig

ure

at t

he r

ight

to

expl

ain

your

ans

wer

.

Yes;

(PQ

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(PB)

impl

ies

that

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! PP QB !.

Sinc

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th "

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to

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E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-2

7-2



Stu

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____

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____

____

____

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____

____

____

PERI

OD

____

_

7-3

7-3

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Lesson 7-3

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at t

he

hyp

oten

use

is

twic

e th

e sh

orte

r le

g an

d t

he

lon

ger

leg

is !

3"ti

mes

th

e sh

orte

r le

g.

!M

NQ

is a

30°

-60°

-90°

righ

t tr

iang

le,a

nd t

he le

ngth

of

the

hypo

tenu

se M "

N"is

tw

o ti

mes

the

leng

th o

f th

e sh

orte

r si

de N"

Q".

Usi

ng t

he P

ytha

gore

an T

heor

em,

a2"

(2x)

2#

x2

"4x

2#

x2

"3x

2

a"

!3x

2"

"x!

3"

In a

30°

-60°

-90°

righ

t tr

ian

gle,

the

hyp

oten

use

is

5 ce

nti

met

ers.

Fin

d t

he

len

gth

s of

th

e ot

her

tw

o si

des

of

the

tria

ngl

e.If

the

hyp

oten

use

of a

30°

-60°

-90°

righ

t tr

iang

le is

5 c

enti

met

ers,

then

the

leng

th o

f th

esh

orte

r le

g is

hal

f of

5 o

r 2.

5 ce

ntim

eter

s.T

he le

ngth

of

the

long

er le

g is

!3"

tim

es t

he

leng

th o

f th

e sh

orte

r le

g,or

(2.

5)(!

3")ce

ntim

eter

s.

Fin

d x

and

y.

1.2.

3.

1;0.

5!3"

#0.

98!

3"#

13.9

;16

5.5;

5.5!

3"#

9.5

4.5.

6.

9;18

4!3"

#6.

9;8!

3"#

13.9

10!

3"#

17.3

;10

7.T

he p

erim

eter

of

an e

quila

tera

l tri

angl

e is

32

cent

imet

ers.

Fin

d th

e le

ngth

of

an a

ltit

ude

of t

he t

rian

gle

to t

he n

eare

st t

enth

of

a ce

ntim

eter

.9.

2 cm

8.A

n al

titu

de o

f an

equ

ilate

ral t

rian

gle

is 8

.3 m

eter

s.F

ind

the

peri

met

er o

f th

e tr

iang

le t

oth

e ne

ares

t te

nth

of a

met

er.

28.8

m

xy

60#

20

xy

60#

12

xy 30

#

9 !%3

x

y11

30#

x

y

60# 8

x y30

#

60#

1 2

x

a

NQP

M

2x30#

30#

60#

60#

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Spec

ial R

ight

Tria

ngle

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-3

7-3

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

!M

NP

is an

equ

ilate

ral

trian

gle.

!M

NQ

is a

30°-6

0°-9

0°rig

ht tr

iang

le.



An

swer

s

Skil

ls P

ract

ice

Spec

ial R

ight

Tria

ngle

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-3

7-3

©G

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w-Hi

ll36

5G

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eom

etry

Lesson 7-3

Fin

d x

and

y.

1.2.

3.

12,1

2!3"

64,3

2!3"

6!2",

6!2"

4.5.

6.

8,8!

2"8,

8!3"

45,1

3!2"

For

Exe

rcis

es 7

–9,u

se t

he

figu

re a

t th

e ri

ght.

7.If

a"

11,f

ind

ban

d c.

b"

11!

3";c

"22

8.If

b"

15,f

ind

aan

d c.

a"

5!3";

c"

10!

3"

9.If

c"

9,fi

nd a

and

b.

a"

4.5;

b"

4.5!

3"

For

Exe

rcis

es 1

0 an

d 1

1,u

se t

he

figu

re a

t th

e ri

ght.

10.T

he p

erim

eter

of

the

squa

re is

30

inch

es.F

ind

the

leng

th o

f B"C"

.7.

5 in

.

11.F

ind

the

leng

th o

f th

e di

agon

al B"

D".

7.5!

2"in

.or a

bout

10.

61 in

.

12.T

he p

erim

eter

of

the

equi

late

ral t

rian

gle

is 6

0 m

eter

s.F

ind

the

le

ngth

of

an a

ltit

ude.

10!

3"m

or a

bout

17.

32 m

13.!

GE

Cis

a 3

0°-6

0°-9

0°tr

iang

le w

ith

righ

t an

gle

at E

,and

E"C"

is

the

long

er le

g.F

ind

the

coor

dina

tes

of G

in Q

uadr

ant

I fo

r E

(1,1

) an

d C

(4,1

).

( 1,1

$!

3")or

abo

ut (1

,2.7

3)

E

FG

D60

#

AB C

D45

#

bA

B Cac

60#

30#

y x# 13

1313

13

y

x60#

16

y

x

45#

8

y

x45

#

12

y

x

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y

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Fin

d x

and

y.

1.2.

3.

,25

!3",

5013

,13!

3"

4.5.

6.

45,1

4!2"

3.5!

3",7

;11!

2"

For

Exe

rcis

es 7

–9,u

se t

he

figu

re a

t th

e ri

ght.

7.If

a"

4!3",

find

ban

d c.

b"

12,c

"8!

3"

8.If

x"

3!3",

find

aan

d C

D.

a"

6!3",

CD"

9

9.If

a"

4,fi

nd C

D,b

,and

y.

CD "

2!3",

b"

4!3",

y"

6

10.T

he p

erim

eter

of

an e

quila

tera

l tri

angl

e is

39

cent

imet

ers.

Fin

d th

e le

ngth

of

an a

ltit

ude

of t

he t

rian

gle.

6.5!

3"in

.or a

bout

11.

26 in

.

11.!

MIP

is a

30°

-60°

-90°

tria

ngle

wit

h ri

ght

angl

e at

I,a

nd I"

P"th

e lo

nger

leg.

Fin

d th

eco

ordi

nate

s of

Min

Qua

dran

t I

for

I(3,

3) a

nd P

(12,

3).

( 3,3

$3!

3")or

abo

ut (3

,8.1

9)

12.!

TJK

is a

45°

-45°

-90°

tria

ngle

wit

h ri

ght

angl

e at

J.F

ind

the

coor

dina

tes

of T

inQ

uadr

ant

II f

or J

(#2,

#3)

and

K(3

,#3)

.

('2,

2)

13.B

OTA

NIC

AL

GA

RD

ENS

One

of

the

disp

lays

at

a bo

tani

cal g

arde

n is

an

herb

gar

den

plan

ted

in t

he s

hape

of

a sq

uare

.The

squ

are

mea

sure

s 6

yard

s on

eac

h si

de.V

isit

ors

can

view

the

her

bs f

rom

adi

agon

al p

athw

ay t

hrou

gh t

he g

arde

n.H

ow lo

ng is

the

pat

hway

?

6!2"

yd o

r abo

ut 8

.48

yd

6 yd

6 yd

6 yd

6 yd

bA

B C

Da

x

y60

#

30#

c

11!

2"!

2

x45

#

11

y60

#3.

5

xy

x#y

28

9!2"

!2

9!2"

!2

y

x30

#

26y

x2560

#

yx

45#

9Pra

ctic

e (A

vera

ge)

Spec

ial R

ight

Tria

ngle

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-3

7-3



Rea

din

g t

o L

earn

Math

emati

csSp

ecia

l Tria

ngle

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-3

7-3

©G

lenc

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cGra

w-Hi

ll36

7G

lenc

oe G

eom

etry

Lesson 7-3

Pre-

Act

ivit

yH

ow i

s tr

ian

gle

tili

ng

use

d i

n w

allp

aper

des

ign

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

3 at

the

top

of

page

357

in y

our

text

book

.•

How

can

you

mos

t co

mpl

etel

y de

scri

be t

he la

rger

tri

angl

e an

d th

e tw

osm

alle

r tr

iang

les

in t

ile 1

5?Sa

mpl

e an

swer

:The

larg

er tr

iang

le is

an is

osce

les

obtu

se tr

iang

le.T

he tw

o sm

alle

r tria

ngle

s ar

eco

ngru

ent s

cale

ne ri

ght t

riang

les.

•H

ow c

an y

ou m

ost

com

plet

ely

desc

ribe

the

larg

er t

rian

gle

and

the

two

smal

ler

tria

ngle

s in

tile

16?

(In

clud

e an

gle

mea

sure

s in

des

crib

ing

all t

hetr

iang

les.

)Sa

mpl

e an

swer

:The

larg

er tr

iang

le is

equ

ilate

ral,

soea

ch o

f its

ang

le m

easu

res

is 6

0.Th

e tw

o sm

alle

r tria

ngle

sar

e co

ngru

ent r

ight

tria

ngle

s in

whi

ch th

e an

gle

mea

sure

sar

e 30

,60,

and

90.

Rea

din

g t

he

Less

on

1.Su

pply

the

cor

rect

num

ber

or n

umbe

rs t

o co

mpl

ete

each

sta

tem

ent.

a.In

a 4

5°-4

5°-9

0°tr

iang

le,t

o fi

nd t

he le

ngth

of

the

hypo

tenu

se,m

ulti

ply

the

leng

th o

f a

leg

by

.

b.In

a 3

0°-6

0°-9

0°tr

iang

le,t

o fi

nd t

he le

ngth

of

the

hypo

tenu

se,m

ulti

ply

the

leng

th o

f

the

shor

ter

leg

by

.

c.In

a 3

0°-6

0°-9

0°tr

iang

le,t

he lo

nger

leg

is o

ppos

ite

the

angl

e w

ith

a m

easu

re o

f .

d.In

a 3

0°-6

0°-9

0°tr

iang

le,t

o fi

nd t

he le

ngth

of

the

long

er le

g,m

ulti

ply

the

leng

th o

f

the

shor

ter

leg

by

.

e.In

an

isos

cele

s ri

ght

tria

ngle

,eac

h le

g is

opp

osit

e an

ang

le w

ith

a m

easu

re o

f .

f.In

a 3

0°-6

0°-9

0°tr

iang

le,t

o fi

nd t

he le

ngth

of

the

shor

ter

leg,

divi

de t

he le

ngth

of

the

long

er le

g by

.

g.In

30°

-60°

-90°

tria

ngle

,to

find

the

leng

th o

f th

e lo

nger

leg,

divi

de t

he le

ngth

of

the

hypo

tenu

se b

y an

d m

ulti

ply

the

resu

lt b

y .

h.

To f

ind

the

leng

th o

f a

side

of

a sq

uare

,div

ide

the

leng

th o

f th

e di

agon

al b

y .

2.In

dica

te w

heth

er e

ach

stat

emen

t is

alw

ays,

som

etim

es,o

r ne

ver

true

.a.

The

leng

ths

of t

he t

hree

sid

es o

f an

isos

cele

s tr

iang

le s

atis

fy t

he P

ytha

gore

anT

heor

em.

som

etim

esb.

The

leng

ths

of t

he s

ides

of

a 30

°-60

°-90

°tr

iang

le f

orm

a P

ytha

gore

an t

ripl

e.ne

ver

c.T

he le

ngth

s of

all

thre

e si

des

of a

30°

-60°

-90°

tria

ngle

are

pos

itiv

e in

tege

rs.

neve

r

Hel

pin

g Y

ou

Rem

emb

er3.

Som

e st

uden

ts f

ind

it e

asie

r to

rem

embe

r m

athe

mat

ical

con

cept

s in

ter

ms

of s

peci

fic

num

bers

rat

her

than

var

iabl

es.H

ow c

an y

ou u

se s

peci

fic

num

bers

to

help

you

rem

embe

rth

e re

lati

onsh

ip b

etw

een

the

leng

ths

of t

he t

hree

sid

es in

a 3

0°-6

0°-9

0°tr

iang

le?

Sam

ple

answ

er:D

raw

a 3

0#-6

0#-9

0#tri

angl

e.La

bel t

he le

ngth

of t

hesh

orte

r leg

as

1.Th

en th

e le

ngth

of t

he h

ypot

enus

e is

2,a

nd th

e le

ngth

of

the

long

er le

g is

!3".

Just

rem

embe

r:1,

2,!

3".

!2"

!3"

2

!3"

45!

3"

602

!2"

©G

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8G

lenc

oe G

eom

etry

Cons

truct

ing

Valu

es o

f Squ

are

Root

sT

he d

iagr

am a

t th

e ri

ght

show

s a

righ

t is

osce

les

tria

ngle

wit

h tw

o le

gs o

f le

ngth

1 in

ch.B

y th

e P

ytha

gore

an T

heor

em,t

he le

ngth

of

the

hyp

oten

use

is !

2"in

ches

.By

cons

truc

ting

an

adja

cent

rig

ht

tria

ngle

wit

h le

gs o

f !2"

inch

es a

nd 1

inch

,you

can

cre

ate

a se

gmen

t of

leng

th !

3".

By

cont

inui

ng t

his

proc

ess

as s

how

n be

low

,you

can

con

stru

ct a

“w

heel

”of

squ

are

root

s.T

his

whe

el is

cal

led

the

“Whe

el o

f The

odor

us”

afte

r a

Gre

ek p

hilo

soph

er w

ho li

ved

abou

t 40

0 B.C

.

Con

tinu

e co

nstr

ucti

ng t

he w

heel

unt

il yo

u m

ake

a se

gmen

t of

leng

th !

18".

!%

1

1

1

3!

%

2

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-3

7-3

1

1

11

1

1!%2

!%3

!%5

!%6

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!"10

!"11

!"12

!"13

!"14

!"15

!"17!"18

!"16

" 4

!%4 "

2

!%9 "

3



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Trig

onom

etry

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-4

7-4

©G

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9G

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Lesson 7-4

Trig

on

om

etri

c R

atio

sT

he r

atio

of

the

leng

ths

of t

wo

side

s of

a r

ight

tr

iang

le is

cal

led

a tr

igon

omet

ric

rati

o.T

he t

hree

mos

t co

mm

on r

atio

s ar

e si

ne,

cosi

ne,

and

tan

gen

t,w

hich

are

abb

revi

ated

sin

,cos

,and

tan

,re

spec

tive

ly.

sin

R"!le

g hyop pp oto es nit ue s# eR

!co

s R

"ta

n R

"

"!r t!

"!s t!

"!r s!

Fin

d s

in A

,cos

A,a

nd

tan

A.E

xpre

ss e

ach

rat

io a

s

a d

ecim

al t

o th

e n

eare

st t

hou

san

dth

.

sin

A"!o hp yp po os ti et ne ul se eg

!co

s A

"!a hd yj pa oc te en nt ul se eg

!ta

n A

"! aop dp jao cs ei nte t

l le eg g!

"!B A

BC !"

! AABC !

"!B A

CC !

"! 15 3!

"!1 12 3!

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#0.

385

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417

Fin

d t

he

ind

icat

ed t

rigo

nom

etri

c ra

tio

as a

fra

ctio

n

and

as

a d

ecim

al.I

f n

eces

sary

,rou

nd

to

the

nea

rest

te

n-t

hou

san

dth

.

1.si

n A

2.ta

n B

!1 15 7!;0

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4! 18 5!

;0.5

333

3.co

s A

4.co

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! 18 7!;0

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5.si

n D

6.ta

n E

!4 5! ;0.

8!3 4! ;

0.75

7.co

s E

8.co

s D

!4 5! ;0.

8!3 5! ;

0.6

16

1620

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3430 CB

AD

FE

12135 CB

A

leg

oppo

site

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!!

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g ad

jace

nt t

o #

Rle

g ad

jace

nt t

o #

R!

!!

hypo

tenu

se

str TS

R

Exer

cises

Exer

cises

Exam

ple

Exam

ple

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oe G

eom

etry

Use

Tri

go

no

met

ric

Rat

ios

In a

rig

ht t

rian

gle,

if y

ou k

now

the

mea

sure

s of

tw

o si

des

or if

you

kno

w t

he m

easu

res

of o

ne s

ide

and

an a

cute

ang

le,t

hen

you

can

use

trig

onom

etri

cra

tios

to

find

the

mea

sure

s of

the

mis

sing

sid

es o

r an

gles

of

the

tria

ngle

.

Fin

d x

,y,a

nd

z.R

oun

d e

ach

mea

sure

to

the

nea

rest

w

hol

e n

um

ber.

1858

#

x#C

Byz

A

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Trig

onom

etry

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-4

7-4

a.F

ind

x.

x%

58"

90x

"32

b.F

ind

y.

tan

A"

! 1y 8!

tan

58°"

! 1y 8!

y"

18 t

an 5

8°y

#29

c.F

ind

z.

cos

A"

!1 z8 !

cos

58°

"!1 z8 !

zco

s 58

°"

18

z"

! cos18

58°

!

z#

34

Exer

cises

Exer

cises

Fin

d x

.Rou

nd

to

the

nea

rest

ten

th.

1.2.

17.0

48.6

3.4.

22.6

76.0

5.6.

24.9

34.2

1564

#x

1640

#

x

4

1x#

12

5x#

1216 x#

3228

#x

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Trig

onom

etry

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-4

7-4

©G

lenc

oe/M

cGra

w-Hi

ll37

1G

lenc

oe G

eom

etry

Lesson 7-4

Use

!R

ST

to f

ind

sin

R,c

os R

,tan

R,s

in S

,cos

S,a

nd

tan

S.

Exp

ress

eac

h r

atio

as

a fr

acti

on a

nd

as

a d

ecim

al t

o th

e n

eare

st h

un

dre

dth

.

1.r

"16

,s"

30,t

"34

2.r

"10

,s"

24,t

"26

sin

R"

!1 36 4!#

0.47

;si

n R

"!1 20 6!

#0.

38;

cos

R"

!3 30 4!#

0.88

;co

s R

"!2 24 6!

#0.

92;

tan

R"

!1 36 0!#

0.53

;ta

n R

"!1 20 4!

#0.

42;

sin

S"

!3 30 4!#

0.88

;si

n S

"!2 24 6!

#0.

92;

cos

S"

!1 36 4!#

0.47

;co

s S

"!1 20 6!

#0.

38;

tan

S"

!3 10 6!#

1.88

tan

S"

!2 14 0!"

2.4

Use

a c

alcu

lato

r to

fin

d e

ach

val

ue.

Rou

nd

to

the

nea

rest

ten

-th

ousa

nd

th.

3.si

n 5

0.08

724.

tan

230.

4245

5.co

s 61

0.48

48

6.si

n 75

.80.

9694

7.ta

n 17

.30.

3115

8.co

s 52

.90.

6032

Use

th

e fi

gure

to

fin

d e

ach

tri

gon

omet

ric

rati

o.E

xpre

ss

answ

ers

as a

fra

ctio

n a

nd

as

a d

ecim

al r

oun

ded

to

the

nea

rest

ten

-th

ousa

nd

th.

9.ta

n C

10.s

in A

11.c

os C

! 49 0!#

0.22

50!4 40 1!

#0.

9756

!4 40 1!#

0.97

56

Fin

d t

he

mea

sure

of

each

acu

te a

ngl

e to

th

e n

eare

st t

enth

of

a d

egre

e.

12.s

in B

"0.

2985

17.4

13.t

an A

"0.

4168

22.6

14.c

os R

"0.

8443

32.4

15.t

an C

"0.

3894

21.3

16.c

os B

"0.

7329

42.9

17.s

in A

"0.

1176

6.8

Fin

d x

.Rou

nd

to

the

nea

rest

ten

th.

18.

19.

20.

28.8

73.5

15.9

19

x

33#

UL

S

27

x#8

BAC

27

x#13 B

A

C

41

409

B

AC

sR

S Trt

©G

lenc

oe/M

cGra

w-Hi

ll37

2G

lenc

oe G

eom

etry

Use

!L

MN

to f

ind

sin

L,c

os L

,tan

L,s

in M

,cos

M,a

nd

tan

M.

Exp

ress

eac

h r

atio

as

a fr

acti

on a

nd

as

a d

ecim

al t

o th

e n

eare

st h

un

dre

dth

.

1.!

"15

,m"

36,n

"39

2.!

"12

,m"

12!

3",n

"24

sin

L"

!1 35 9!#

0.38

;si

n L

"!1 22 4!

"0.

50;

cos

L"

!3 36 9!#

0.92

;co

s L

"#

0.87

;

tan

L"

!1 35 6!#

0.42

;ta

n L

"#

0.58

;

sin

M"

!3 36 9!#

0.92

;si

n M

"#

0.87

;

cos

M"

!1 35 9!#

0.38

;co

s M

"!1 22 4!

"0.

50;

tan

M"

!3 16 5!"

2.4

tan

M"

#1.

73

Use

a c

alcu

lato

r to

fin

d e

ach

val

ue.

Rou

nd

to

the

nea

rest

ten

-th

ousa

nd

th.

3.si

n 92

.40.

9991

4.ta

n 27

.50.

5206

5.co

s 64

.80.

4258

Use

th

e fi

gure

to

fin

d e

ach

tri

gon

omet

ric

rati

o.E

xpre

ss

answ

ers

as a

fra

ctio

n a

nd

as

a d

ecim

al r

oun

ded

to

the

nea

rest

ten

-th

ousa

nd

th.

6.co

s A

7.ta

n B

8.si

n A

#0.

9487

!3 1!"

3.00

00#

0.31

62

Fin

d t

he

mea

sure

of

each

acu

te a

ngl

e to

th

e n

eare

st t

enth

of

a d

egre

e.

9.si

n B

"0.

7823

51.5

10.t

an A

"0.

2356

13.3

11.c

os R

"0.

6401

50.2

Fin

d x

.Rou

nd

to

the

nea

rest

ten

th.

12.

64.4

13.

18.1

14.

24.2

15.G

EOG

RA

PHY

Die

go u

sed

a th

eodo

lite

to m

ap a

reg

ion

of la

nd f

or h

is

clas

s in

geo

mor

phol

ogy.

To d

eter

min

e th

e el

evat

ion

of a

ver

tica

l roc

kfo

rmat

ion,

he m

easu

red

the

dist

ance

fro

m t

he b

ase

of t

he f

orm

atio

n to

hi

s po

siti

on a

nd t

he a

ngle

bet

wee

n th

e gr

ound

and

the

line

of

sigh

t to

th

e to

p of

the

for

mat

ion.

The

dis

tanc

e w

as 4

3 m

eter

s an

d th

e an

gle

was

36

deg

rees

.Wha

t is

the

hei

ght

of t

he f

orm

atio

n to

the

nea

rest

met

er?

31 m

36# 43

m

41#

x

3229

x#9

23

x#11

!10"

!10

3!10"

!10

15

5 !"10

5 CA

B

12!

3"!

12

12!

3"!

2412! 12

!3"

12!

3"!

24

ML

N

Pra

ctic

e (A

vera

ge)

Trig

onom

etry

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-4

7-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csTr

igon

omet

ry

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-4

7-4

©G

lenc

oe/M

cGra

w-Hi

ll37

3G

lenc

oe G

eom

etry

Lesson 7-4

Pre-

Act

ivit

yH

ow c

an s

urv

eyor

s d

eter

min

e an

gle

mea

sure

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

4 at

the

top

of

page

364

in y

our

text

book

.

•W

hy is

it im

port

ant

to d

eter

min

e th

e re

lati

ve p

osit

ions

acc

urat

ely

inna

viga

tion

? (G

ive

two

poss

ible

rea

sons

.)Sa

mpl

e an

swer

s:(1

) To

avoi

d co

llisi

ons

betw

een

ship

s,an

d (2

) to

prev

ent s

hips

from

losi

ng th

eir b

earin

gs a

nd g

ettin

g lo

st a

t sea

.•

Wha

t do

es c

alib

rate

dm

ean?

Sam

ple

answ

er:m

arke

d pr

ecis

ely

tope

rmit

accu

rate

mea

sure

men

ts to

be

mad

eR

ead

ing

th

e Le

sso

n

1.R

efer

to

the

figu

re.W

rite

a r

atio

usi

ng t

he s

ide

leng

ths

in t

he

figu

re t

o re

pres

ent

each

of

the

follo

win

g tr

igon

omet

ric

rati

os.

A.s

in N

! MMNP !

B.c

os N

! MNP N!

C.t

an N

!M NPP !

D.t

an M

! MNPP!

E.s

in M

! MNP N!F.

cos

M! MM

NP !

2.A

ssum

e th

at y

ou e

nter

eac

h of

the

exp

ress

ions

in t

he li

st o

n th

e le

ft in

to y

our

calc

ulat

or.

Mat

ch e

ach

of t

hese

exp

ress

ions

wit

h a

desc

ript

ion

from

the

list

on

the

righ

t to

tel

l wha

tyo

u ar

e fi

ndin

g w

hen

you

ente

r th

is e

xpre

ssio

n.

P

MN

a.si

n 20

vb.

cos

20ii

c.si

n#1

0.8

vid.

tan#

10.

8iii

e.ta

n 20

ivf.

cos#

10.

8i

i.th

e de

gree

mea

sure

of

an a

cute

ang

le w

hose

cos

ine

is 0

.8

ii.

the

rati

o of

the

leng

th o

f th

e le

g ad

jace

nt t

o th

e 20

°an

gle

to t

hele

ngth

of

hypo

tenu

se in

a 2

0°-7

0°-9

0°tr

iang

le

iii.

the

degr

ee m

easu

re o

f an

acu

te a

ngle

in a

rig

ht t

rian

gle

for

whi

ch t

he r

atio

of

the

leng

th o

f th

e op

posi

te le

g to

the

leng

th o

fth

e ad

jace

nt le

g is

0.8

iv.t

he r

atio

of t

he le

ngth

of t

he le

g op

posi

te t

he 2

0°an

gle

to t

hele

ngth

of t

he le

g ad

jace

nt t

o it

in a

20°

-70°

-90°

tria

ngle

v.th

e ra

tio

of t

he le

ngth

of

the

leg

oppo

site

the

20°

angl

e to

the

leng

th o

f hy

pote

nuse

in a

20°

-70°

-90°

tria

ngle

vi.t

he d

egre

e m

easu

re o

f an

acu

te a

ngle

in a

rig

ht t

rian

gle

for

whi

ch t

he r

atio

of

the

leng

th o

f th

e op

posi

te le

g to

the

leng

th o

fth

e hy

pote

nuse

is 0

.8

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an t

he c

oin

cos

ine

help

you

to

rem

embe

r th

e re

lati

onsh

ip b

etw

een

the

sine

s an

dco

sine

s of

the

tw

o ac

ute

angl

es o

f a

righ

t tr

iang

le?

Sam

ple

answ

er:T

he c

oin

cos

ine

com

es fr

om c

ompl

emen

t,as

inco

mpl

emen

tary

angl

es.T

he c

osin

e of

an

acut

e an

gle

is e

qual

to th

e si

neof

its

com

plem

ent.

©G

lenc

oe/M

cGra

w-Hi

ll37

4G

lenc

oe G

eom

etry

Sine

and

Cos

ine

of A

ngle

sT

he f

ollo

win

g di

agra

m c

an b

e us

ed t

o ob

tain

app

roxi

mat

e va

lues

for

the

sin

ean

d co

sine

of

angl

es f

rom

0°

to 9

0°.T

he r

adiu

s of

the

cir

cle

is 1

.So,

the

sine

and

cosi

ne v

alue

s ca

n be

rea

d di

rect

ly f

rom

the

ver

tica

l and

hor

izon

tal a

xes.

Fin

d a

pp

roxi

mat

e va

lues

for

sin

40°

and

cos

40#

.Con

sid

er t

he

tria

ngl

e fo

rmed

by

the

segm

ent

mar

ked

40°

,as

illu

stra

ted

by

the

shad

ed

tria

ngl

e at

rig

ht.

sin

40°

"!a c!

#!0.

164 !or

0.6

4co

s 40

°"

!b c!#

!0.177 !

or 0

.77

1.U

se t

he d

iagr

am a

bove

to

com

plet

e th

e ch

art

of v

alue

s.

2.C

ompa

re t

he s

ine

and

cosi

ne o

f tw

o co

mpl

emen

tary

ang

les

(ang

les

who

se

sum

is 9

0°).

Wha

t do

you

not

ice?

The

sine

of a

n an

gle

is e

qual

to th

e co

sine

of t

he c

ompl

emen

t of

the

angl

e.

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

90°

0°10°

20°

30°

40°

50°

60°

70°

80°

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-4

7-4 x°

0°10

°20

°30

°40

°50

°60

°70

°80

°90

°

sin x

°0

0.17

0.34

0.5

0.64

0.77

0.87

0.94

0.98

1co

s x°

10.

980.

940.

870.

770.

640.

50.

340.

170

1 0

40°

0.64

c "

1 u

nit

x°b

" c

os x

°0.

771

a "

sin

x°

Exam

ple

Exam

ple



Stu

dy

Gu

ide

and I

nte

rven

tion

Angl

es o

f Ele

vatio

n an

d De

pres

sion

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-5

7-5

©G

lenc

oe/M

cGra

w-Hi

ll37

5G

lenc

oe G

eom

etry

Lesson 7-5

An

gle

s o

f El

evat

ion

Man

y re

al-w

orld

pro

blem

s th

at in

volv

e lo

okin

g up

to

an o

bjec

t ca

n be

des

crib

ed in

ter

ms

of a

n an

gle

of

elev

atio

n,w

hich

is t

he a

ngle

bet

wee

n an

obs

erve

r’s li

ne o

f si

ght

and

a ho

rizo

ntal

line

.

Th

e an

gle

of e

leva

tion

fro

m p

oin

t A

to t

he

top

of

a c

liff

is

34°.

If p

oin

t A

is 1

000

feet

fro

m t

he

base

of

the

clif

f,h

ow h

igh

is

the

clif

f?L

et x

"th

e he

ight

of

the

clif

f.

tan

34°

"! 10

x 00!ta

n "

!o ap dp jao cs eit ne t!

1000

(tan

34°

)"

xM

ultip

ly ea

ch s

ide

by 1

000.

674.

5"

xUs

e a

calcu

lato

r.

The

hei

ght

of t

he c

liff

is a

bout

674

.5 f

eet.

Sol

ve e

ach

pro

blem

.Rou

nd

mea

sure

s of

seg

men

ts t

o th

e n

eare

st w

hol

e n

um

ber

and

an

gles

to

the

nea

rest

deg

ree.

1.T

he a

ngle

of

elev

atio

n fr

om p

oint

Ato

the

top

of

a hi

ll is

49°

.If

poi

nt A

is 4

00 f

eet

from

the

bas

e of

the

hill

,how

hig

h is

th

e hi

ll?

460

ft

2.F

ind

the

angl

e of

ele

vati

on o

f th

e su

n w

hen

a 12

.5-m

eter

-tal

l te

leph

one

pole

cas

ts a

18-

met

er-l

ong

shad

ow.

35°

3.A

ladd

er le

anin

g ag

ains

t a

build

ing

mak

es a

n an

gle

of 7

8°w

ith

the

grou

nd.T

he f

oot

of t

he la

dder

is 5

fee

t fr

om t

he

build

ing.

How

long

is t

he la

dder

?

24 ft

4.A

per

son

who

se e

yes

are

5 fe

et a

bove

the

gro

und

is s

tand

ing

on t

he r

unw

ay o

f an

air

port

100

fee

t fr

om t

he c

ontr

ol t

ower

.T

hat

pers

on o

bser

ves

an a

ir t

raff

ic c

ontr

olle

r at

the

win

dow

of

the

132

-foo

t to

wer

.Wha

t is

the

ang

le o

f el

evat

ion?

52°

?5

ft10

0 ft

132

ft

78#

5 ft

?

18 m

12.5

msun

?

✹

400

ft

?

49#

A

?

1000

ft34

#A

x

angl

e of

elev

atio

n

line o

f sigh

t

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-Hi

ll37

6G

lenc

oe G

eom

etry

An

gle

s o

f D

epre

ssio

nW

hen

an o

bser

ver

is lo

okin

g do

wn,

the

angl

e of

dep

ress

ion

is t

he a

ngle

bet

wee

n th

e ob

serv

er’s

line

of

sigh

t an

d a

hori

zont

al li

ne.

Th

e an

gle

of d

epre

ssio

n f

rom

th

e to

p o

f an

80

-foo

t bu

ild

ing

to p

oin

t A

on t

he

grou

nd

is

42°.

How

far

is

th

e fo

ot o

f th

e bu

ild

ing

from

poi

nt

A?

Let

x"

the

dist

ance

fro

m p

oint

Ato

the

foo

t of

the

bui

ldin

g.Si

nce

the

hori

zont

al li

ne is

par

alle

l to

the

grou

nd,t

he a

ngle

of

depr

essi

on#

DB

Ais

con

grue

nt t

o #

BA

C.

tan

42°

"!8 x0 !

tan

"!o ap dp jao cs eit ne t

!

x(ta

n 42

°)"

80M

ultip

ly ea

ch s

ide

by x

.

x"

! tan80

42°

!Di

vide

each

sid

e by

tan

42°.

x#

88.8

Use

a ca

lcula

tor.

Poin

t A

is a

bout

89

feet

fro

m t

he b

ase

of t

he b

uild

ing.

Sol

ve e

ach

pro

blem

.Rou

nd

mea

sure

s of

seg

men

ts t

o th

e n

eare

st w

hol

e n

um

ber

and

an

gles

to

the

nea

rest

deg

ree.

1.T

he a

ngle

of

depr

essi

on f

rom

the

top

of

a sh

eer

clif

f to

po

int

Aon

the

gro

und

is 3

5°.I

f po

int

Ais

280

fee

t fr

om

the

base

of

the

clif

f,ho

w t

all i

s th

e cl

iff?

196

ft

2.T

he a

ngle

of

depr

essi

on f

rom

a b

allo

on o

n a

75-f

oot

stri

ng t

o a

pers

on o

n th

e gr

ound

is 3

6°.H

ow h

igh

is

the

ballo

on?

44 ft

3.A

ski

run

is 1

000

yard

s lo

ng w

ith

a ve

rtic

al d

rop

of

208

yard

s.F

ind

the

angl

e of

dep

ress

ion

from

the

top

of

the

ski

run

to

the

bott

om.

12°

4.F

rom

the

top

of

a 12

0-fo

ot-h

igh

tow

er,a

n ai

r tr

affi

c co

ntro

ller

obse

rves

an

airp

lane

on

the

runw

ay a

t an

an

gle

of d

epre

ssio

n of

19°

.How

far

fro

m t

he b

ase

of t

heto

wer

is t

he a

irpl

ane?

349

ft

120

ft

?

19#

208

yd

?

1000

yd

36#

75 ft

?

A

35#

280

ft

?

ACB

D

x42

#

angl

e of

depr

essi

on

horiz

onta

l

80 ftY

line o

f sigh

t

horiz

onta

lan

gle

ofde

pres

sion

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Angl

es o

f Ele

vatio

n an

d De

pres

sion

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-5

7-5

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Angl

es o

f Ele

vatio

n an

d De

pres

sion

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-5

7-5

©G

lenc

oe/M

cGra

w-Hi

ll37

7G

lenc

oe G

eom

etry

Lesson 7-5

Nam

e th

e an

gle

of d

epre

ssio

n o

r an

gle

of e

leva

tion

in

eac

h f

igu

re.

1.2.

"FL

S;"

TSL

"RT

W;"

SWT

3.4.

"DC

B;"

ABC

"W

ZP;"

RPZ

5.M

OU

NTA

IN B

IKIN

GO

n a

mou

ntai

n bi

ke t

rip

alon

g th

e G

emin

i Bri

dges

Tra

il in

Moa

b,U

tah,

Nab

uko

stop

ped

on t

he c

anyo

n fl

oor

to g

et a

goo

d vi

ew o

f th

e tw

in s

ands

tone

brid

ges.

Nab

uko

is s

tand

ing

abou

t 60

met

ers

from

the

bas

e of

the

can

yon

clif

f,an

d th

ena

tura

l arc

h br

idge

s ar

e ab

out

100

met

ers

up t

he c

anyo

n w

all.

If h

er li

ne o

f si

ght

is f

ive

feet

abo

ve t

he g

roun

d,w

hat

is t

he a

ngle

of

elev

atio

n to

the

top

of

the

brid

ges?

Rou

nd t

oth

e ne

ares

t te

nth

degr

ee.

abou

t 57.

7#

6.SH

AD

OW

SSu

ppos

e th

e su

n ca

sts

a sh

adow

off

a 3

5-fo

ot b

uild

ing.

If t

he a

ngle

of

elev

atio

n to

the

sun

is 6

0°,h

ow lo

ng is

the

sha

dow

to

the

nea

rest

ten

th o

f a

foot

?ab

out 2

0.2

ft

7.B

ALL

OO

NIN

GF

rom

her

pos

itio

n in

a h

ot-a

ir b

allo

on,A

ngie

can

see

her

car

par

ked

in a

fiel

d.If

the

ang

le o

f de

pres

sion

is 8

°an

d A

ngie

is 3

8 m

eter

s ab

ove

the

grou

nd,w

hat

isth

e st

raig

ht-l

ine

dist

ance

fro

m A

ngie

to

her

car?

Rou

nd t

o th

e ne

ares

t w

hole

met

er.

abou

t 273

m

8.IN

DIR

ECT

MEA

SUR

EMEN

TK

yle

is a

t th

e en

d of

a p

ier

30 f

eet

abov

e th

e oc

ean.

His

eye

leve

l is

3 fe

et a

bove

the

pie

r.H

e is

usi

ng b

inoc

ular

s to

w

atch

a w

hale

sur

face

.If

the

angl

e of

dep

ress

ion

of t

he w

hale

is 2

0°,h

ow f

ar is

the

wha

le f

rom

K

yle’

s bi

nocu

lars

? R

ound

to

the

near

est

tent

h fo

ot.

abou

t 96.

5 ft

wha

lew

ater

leve

l

20#

Kyle

’s ey

es

pier

3 ft

30 ft

60# ?

35 ft

Z

PW

R

D

AC

B

T

WR

S

F

T

L

S

©G

lenc

oe/M

cGra

w-Hi

ll37

8G

lenc

oe G

eom

etry

Nam

e th

e an

gle

of d

epre

ssio

n o

r an

gle

of e

leva

tion

in

eac

h f

igu

re.

1.2.

"TR

Z;"

YZR

"PR

M;"

LMR

3.W

ATE

R T

OW

ERS

A s

tude

nt c

an s

ee a

wat

er t

ower

fro

m t

he c

lose

st p

oint

of

the

socc

erfi

eld

at S

an L

obos

Hig

h Sc

hool

.The

edg

e of

the

soc

cer

fiel

d is

abo

ut 1

10 f

eet

from

the

wat

er t

ower

and

the

wat

er t

ower

sta

nds

at a

hei

ght

of 3

2.5

feet

.Wha

t is

the

ang

le o

fel

evat

ion

if t

he e

ye le

vel o

f th

e st

uden

t vi

ewin

g th

e to

wer

fro

m t

he e

dge

of t

he s

occe

rfi

eld

is 6

fee

t ab

ove

the

grou

nd?

Rou

nd t

o th

e ne

ares

t te

nth

degr

ee.

abou

t 13.

5#

4.C

ON

STR

UC

TIO

NA

roo

fer

prop

s a

ladd

er a

gain

st a

wal

l so

that

the

top

of

the

ladd

erre

ache

s a

30-f

oot

roof

tha

t ne

eds

repa

ir.I

f th

e an

gle

of e

leva

tion

fro

m t

he b

otto

m o

f th

ela

dder

to

the

roof

is 5

5°,h

ow f

ar is

the

ladd

er f

rom

the

bas

e of

the

wal

l? R

ound

you

ran

swer

to

the

near

est

foot

.

abou

t 21

ft

5.TO

WN

OR

DIN

AN

CES

The

tow

n of

Bel

mon

t re

stri

cts

the

heig

ht

of f

lagp

oles

to

25 f

eet

on a

ny p

rope

rty.

Lin

dsay

wan

ts t

o de

term

ine

whe

ther

her

sch

ool i

s in

com

plia

nce

wit

h th

e re

gula

tion

.Her

eye

le

vel i

s 5.

5 fe

et f

rom

the

gro

und

and

she

stan

ds 3

6 fe

et f

rom

the

flag

pole

.If

the

angl

e of

ele

vati

on is

abo

ut 2

5°,w

hat

is t

he h

eigh

t of

the

fla

gpol

e to

the

nea

rest

ten

th f

oot?

abou

t 22.

3 ft

6.G

EOG

RA

PHY

Step

han

is s

tand

ing

on a

mes

a at

the

Pai

nted

Des

ert.

The

ele

vati

on o

fth

e m

esa

is a

bout

138

0 m

eter

s an

d St

epha

n’s

eye

leve

l is

1.8

met

ers

abov

e gr

ound

.If

Step

han

can

see

a ba

nd o

f m

ulti

colo

red

shal

e at

the

bot

tom

and

the

ang

le o

f de

pres

sion

is 2

9°,a

bout

how

far

is t

he b

and

of s

hale

fro

m h

is e

yes?

Rou

nd t

o th

e ne

ares

t m

eter

.

abou

t 285

0 m

7.IN

DIR

ECT

MEA

SUR

EMEN

TM

r.D

omin

guez

is s

tand

ing

on a

40-

foot

oce

an b

luff

nea

r hi

s ho

me.

He

can

see

his

two

dogs

on

the

beac

h be

low

.If

his

line

of s

ight

is 6

fee

t ab

ove

the

grou

nd a

nd t

he a

ngle

s of

dep

ress

ion

to h

is d

ogs

are

34°

and

48°,

how

far

apa

rt a

re t

he d

ogs

to t

he n

eare

st f

oot?

abou

t 27

ft48

#34

#

40 ft

6 ft

Mr.

Dom

ingu

ez

bluf

f

25#

5.5

ft36

ft

x

R

M

P

L

T

YR

Z

Pra

ctic

e (A

vera

ge)

Angl

es o

f Ele

vatio

n an

d De

pres

sion

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-5

7-5



Rea

din

g t

o L

earn

Math

emati

csAn

gles

of E

leva

tion

and

Depr

essi

on

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-5

7-5

©G

lenc

oe/M

cGra

w-Hi

ll37

9G

lenc

oe G

eom

etry

Lesson 7-5

Pre-

Act

ivit

yH

ow d

o ai

rlin

e p

ilot

s u

se a

ngl

es o

f el

evat

ion

an

d d

epre

ssio

n?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

5 at

the

top

of

page

371

in y

our

text

book

.

Wha

t do

es t

he a

ngle

mea

sure

tel

l the

pilo

t?Sa

mpl

e an

swer

:how

stee

p he

r asc

ent m

ust b

e to

cle

ar th

e pe

ak

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.T

he t

wo

obse

rver

s ar

e lo

okin

g at

on

e an

othe

r.Se

lect

the

cor

rect

cho

ice

for

each

que

stio

n.

a.W

hat

is t

he li

ne o

f si

ght?

iii(i

) lin

e R

S(i

i) li

ne S

T(i

ii) li

ne R

T(i

v) li

ne T

U

b.W

hat

is t

he a

ngle

of

elev

atio

n?ii

(i) #

RS

T(i

i) #

SR

T(i

ii) #

RT

S(i

v) #

UT

R

c.W

hat

is t

he a

ngle

of

depr

essi

on?

iv(i

) #R

ST

(ii)

#S

RT

(iii)

#R

TS

(iv)

#U

TR

d.H

ow a

re t

he a

ngle

of

elev

atio

n an

d th

e an

gle

of d

epre

ssio

n re

late

d?ii

(i)

The

y ar

e co

mpl

emen

tary

.(i

i)T

hey

are

cong

ruen

t.(i

ii)T

hey

are

supp

lem

enta

ry.

(iv)

The

ang

le o

f el

evat

ion

is la

rger

tha

n th

e an

gle

of d

epre

ssio

n.

e.W

hich

pos

tula

te o

r th

eore

m t

hat

you

lear

ned

in C

hapt

er 3

sup

port

s yo

ur a

nsw

er f

orpa

rt c

?iv

(i)

Cor

resp

ondi

ng A

ngle

s Po

stul

ate

(ii)

Alt

erna

te E

xter

ior

Ang

les

The

orem

(iii)

Con

secu

tive

Int

erio

r A

ngle

s T

heor

em(i

v)A

lter

nate

Int

erio

r A

ngle

s T

heor

em

2.A

stu

dent

say

s th

at t

he a

ngle

of

elev

atio

n fr

om h

is e

ye t

o th

e to

p of

a f

lagp

ole

is 1

35°.

Wha

t is

wro

ng w

ith

the

stud

ent’s

sta

tem

ent?

An a

ngle

of e

leva

tion

cann

ot b

e ob

tuse

.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o so

meo

ne e

lse.

Supp

ose

a cl

assm

ate

finds

it d

iffic

ult

to d

isti

ngui

sh b

etw

een

angl

es o

f ele

vati

on a

nd a

ngle

s of

dep

ress

ion.

Wha

tar

e so

me

hint

s yo

u ca

n gi

ve h

er t

o he

lp h

er g

et it

rig

ht e

very

tim

e?Sa

mpl

e an

swer

s:(1

) The

ang

le o

f dep

ress

ion

and

the

angl

e of

ele

vatio

n ar

e bo

th m

easu

red

betw

een

the

horiz

onta

l and

the

line

of s

ight

.(2)

The

ang

le o

f dep

ress

ion

is a

lway

s co

ngru

ent t

o th

e an

gle

of e

leva

tion

in th

e sa

me

diag

ram

.(3

) Ass

ocia

te th

e w

ord

elev

atio

nw

ith th

e w

ord

upan

d th

e w

ord

depr

essi

onw

ith th

e w

ord

dow

n.

STob

serv

er a

tto

p of

bui

ldin

g

obse

rver

on g

roun

dR

U

©G

lenc

oe/M

cGra

w-Hi

ll38

0G

lenc

oe G

eom

etry

Read

ing

Mat

hem

atic

sT

he t

hree

mos

t co

mm

on t

rigo

nom

etri

c ra

tios

are

si

ne,

cosi

ne,

and

tan

gen

t.T

hree

oth

er r

atio

s ar

e th

eco

seca

nt,

seca

nt,

and

cota

nge

nt.

The

cha

rt b

elow

sh

ows

abbr

evia

tion

s an

d de

fini

tion

s fo

r al

l six

rat

ios.

Ref

er t

o th

e tr

iang

le a

t th

e ri

ght.

Use

th

e ab

brev

iati

ons

to r

ewri

te e

ach

sta

tem

ent

as a

n e

quat

ion

.

1.T

he s

ecan

t of

ang

le A

is e

qual

to

1 di

vide

d by

the

cos

ine

of a

ngle

A.

sec

A"

! co1 s

A!

2.T

he c

osec

ant

of a

ngle

A is

equ

al t

o 1

divi

ded

by t

he s

ine

of a

ngle

A.

csc

A"

! sin1

A!

3.T

he c

otan

gent

of

angl

e A

is e

qual

to

1 di

vide

d by

the

tan

gent

of

angl

e A

.co

t A"

! tan1

A!

4.T

he c

osec

ant

of a

ngle

A m

ulti

plie

d by

the

sin

e of

ang

le A

is e

qual

to

1.cs

c A

sin

A"

1

5.T

he s

ecan

t of

ang

le A

mul

tipl

ied

by t

he c

osin

e of

ang

le A

is e

qual

to

1.se

c A

cos

A"

1

6.T

he c

otan

gent

of

angl

e A

tim

es t

he t

ange

nt o

f an

gle

Ais

equ

al t

o 1.

cot A

tan

A"

1

Use

th

e tr

ian

gle

at r

igh

t.W

rite

eac

h r

atio

.

7.se

c R

! st !8.

csc

R! rt !

9.co

t R

!s r!

10.s

ec S

! rt !11

.cs

c S

! st !12

.co

t S

! sr !

13.I

f si

n x°

"0.

289,

find

the

val

ue o

f cs

c x°.

#3.

46

14.I

f ta

n x°

"1.

376,

find

the

val

ue o

f co

t x°.

#0.

727

R TS

ts

r

A

ca

b

B C

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-5

7-5

Abbr

evia

tion

Read

as:

Ratio

sin A

the

sine

of #

A"

!a c!

cos

Ath

e co

sine

of #

A"

!b c!

tan

Ath

e ta

ngen

t of #

A"

!a b!

csc

Ath

e co

seca

nt o

f #A

"! ac !

sec

Ath

e se

cant

of #

A"

! bc !

cot A

the

cota

ngen

t of #

A"

!b a!le

gad

jace

nt to

#A

!!

!le

gop

posit

e#

A

hypo

tenu

se!

!!

leg

adja

cent

to#

A

hypo

tenu

se!

!le

gop

posit

e#

A

leg

oppo

site

#A

!!

!le

gad

jace

nt to

#A

leg

adja

cent

to#

A!

!!

hypo

tenu

se

leg

oppo

site

#A

!!

hypo

tenu

se



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

The

Law

of S

ines

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-6

7-6

©G

lenc

oe/M

cGra

w-Hi

ll38

1G

lenc

oe G

eom

etry

Lesson 7-6

The

Law

of

Sin

esIn

any

tri

angl

e,th

ere

is a

spe

cial

rel

atio

nshi

p be

twee

n th

e an

gles

of

the

tria

ngle

and

the

leng

ths

of t

he s

ides

opp

osit

e th

e an

gles

.

Law

of S

ines

!sinaA !

"!sin

bB !"

!sincC !

In !

AB

C,f

ind

b.

!sin c

C !"

!sin b

B !La

w of

Sin

es

!sin 30

45°

!"

!sin b74

°!

m#

C"

45, c

"30

, m#

B"

74

bsi

n 45

°"

30 s

in 7

4°Cr

oss

mul

tiply.

b"

!30si

s nin 457 °4°

!Di

vide

each

sid

e by

sin

45°

.

b#

40.8

Use

a ca

lcula

tor.

45#

3074

#

bB

AC

In !

DE

F,f

ind

m"

D.

!sin d

D !"

!sin e

E !La

w of

Sin

es

!si2n 8D !

"!si

n 2458

°!

d"

28, m

#E

"58

, e

"24

24 s

in D

"28

sin

58°

Cros

s m

ultip

ly.

sin

D"

!28s 2in 4

58°

!Di

vide

each

sid

e by

24.

D"

sin#

1 !28

s 2in 458

°!

Use

the

inve

rse

sine.

D#

81.6

°Us

e a

calcu

lato

r.

58#

24

28

E

FD

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

of !

AB

C.R

oun

d a

ngl

e m

easu

res

toth

e n

eare

st d

egre

e an

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

1.If

c"

12,m

#A

"80

,and

m#

C"

40,f

ind

a.18

.4

2.If

b"

20,c

"26

,and

m#

C"

52,f

ind

m#

B.

37

3.If

a"

18,c

"16

,and

m#

A"

84,f

ind

m#

C.

62

4.If

a"

25,m

#A

"72

,and

m#

B"

17,f

ind

b.7.

7

5.If

b"

12,m

#A

"89

,and

m#

B"

80,f

ind

a.

12.2

6.If

a"

30,c

"20

,and

m#

A"

60,f

ind

m#

C.

35

©G

lenc

oe/M

cGra

w-Hi

ll38

2G

lenc

oe G

eom

etry

Use

th

e La

w o

f Si

nes

to

So

lve

Pro

ble

ms

You

can

use

the

Law

of

Sin

esto

sol

veso

me

prob

lem

s th

at in

volv

e tr

iang

les.

Law

of S

ines

Let !

ABC

be a

ny tr

iang

le w

ith a

, b, a

nd c

repr

esen

ting

the

mea

sure

s of

the

sides

opp

osite

th

e an

gles

with

mea

sure

s A,

B, a

nd C

, res

pect

ively.

The

n !sin

aA !"

!sinbB !

"!sin

cC !.

Isos

cele

s !

AB

Ch

as a

bas

e of

24

cen

tim

eter

s an

d a

ve

rtex

an

gle

of 6

8°.F

ind

th

e p

erim

eter

of

the

tria

ngl

e.T

he v

erte

x an

gle

is 6

8°,s

o th

e su

m o

f th

e m

easu

res

of t

he b

ase

angl

es is

11

2 an

d m

#A

"m

#C

"56

.

!sin b

B !"

!sin a

A !La

w of

Sin

es

!sin 24

68°

!"

!sin a56

°!

m#

B"

68, b

"24

, m#

A"

56

asi

n 68

°"

24 s

in 5

6°Cr

oss

mul

tiply.

a"

!24si

s nin 685 °6°

!Di

vide

each

sid

e by

sin

68°

.

#21

.5Us

e a

calcu

lato

r.

The

tri

angl

e is

isos

cele

s,so

c"

21.5

.T

he p

erim

eter

is 2

4 %

21.5

%21

.5 o

r ab

out

67 c

enti

met

ers.

Dra

w a

tri

angl

e to

go

wit

h e

ach

exe

rcis

e an

d m

ark

it

wit

h t

he

give

n i

nfo

rmat

ion

.T

hen

sol

ve t

he

pro

blem

.Rou

nd

an

gle

mea

sure

s to

th

e n

eare

st d

egre

e an

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

1.O

ne s

ide

of a

tri

angu

lar

gard

en is

42.

0 fe

et.T

he a

ngle

s on

eac

h en

d of

thi

s si

de m

easu

re66

°an

d 82

°.F

ind

the

leng

th o

f fe

nce

need

ed t

o en

clos

e th

e ga

rden

.19

2.9

ft

2.T

wo

rada

r st

atio

ns A

and

Bar

e 32

mile

s ap

art.

The

y lo

cate

an

airp

lane

Xat

the

sam

eti

me.

The

thr

ee p

oint

s fo

rm #

XA

B,w

hich

mea

sure

s 46

°,an

d #

XB

A,w

hich

mea

sure

s52

°.H

ow f

ar is

the

air

plan

e fr

om e

ach

stat

ion?

25.5

mi f

rom

A;2

3.2

mi f

rom

B

3.A

civ

il en

gine

er w

ants

to

dete

rmin

e th

e di

stan

ces

from

poi

nts

Aan

d B

to a

n in

acce

ssib

lepo

int

Cin

a r

iver

.#B

AC

mea

sure

s 67

°an

d #

AB

Cm

easu

res

52°.

If p

oint

s A

and

Bar

e82

.0 f

eet

apar

t,fi

nd t

he d

ista

nce

from

Cto

eac

h po

int.

86.3

ft to

poi

nt B

;73.

9 ft

to p

oint

A

4.A

ran

ger

tow

er a

t po

int

Ais

42

kilo

met

ers

nort

h of

a r

ange

r to

wer

at

poin

t B

.A fi

re a

tpo

int

Cis

obs

erve

d fr

om b

oth

tow

ers.

If #

BA

Cm

easu

res

43°

and

#A

BC

mea

sure

s 68

°,w

hich

ran

ger

tow

er is

clo

ser

to t

he f

ire?

How

muc

h cl

oser

?To

wer

Bis

11

km c

lose

r tha

n To

wer

A.

68# b

ca

24B

CA

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

The

Law

of S

ines

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-6

7-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

The

Law

of S

ines

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-6

7-6

©G

lenc

oe/M

cGra

w-Hi

ll38

3G

lenc

oe G

eom

etry

Lesson 7-6

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

from

!A

BC

.Rou

nd

an

gle

mea

sure

sto

th

e n

eare

st t

enth

deg

ree

and

sid

e m

easu

res

to t

he

nea

rest

ten

th.

1.If

m#

A"

35,m

#B

"48

,and

b"

28,f

ind

a.21

.6

2.If

m#

B"

17,m

#C

"46

,and

c"

18,f

ind

b.7.

3

3.If

m#

C"

86,m

#A

"51

,and

a"

38,f

ind

c.48

.8

4.If

a"

17,b

"8,

and

m#

A"

73,f

ind

m#

B.

26.7

5.If

c"

38,b

"34

,and

m#

B"

36,f

ind

m#

C.

41.1

or 1

38.9

6.If

a"

12,c

"20

,and

m#

C"

83,f

ind

m#

A.

36.6

7.If

m#

A"

22,a

"18

,and

m#

B"

104,

find

b.

46.6

Sol

ve e

ach

!P

QR

des

crib

ed b

elow

.Rou

nd

mea

sure

s to

th

e n

eare

st t

enth

.

8.p

"27

,q"

40,m

#P

"33

m"

Q#

53.8

,m"

R#

93.2

,r#

49.5

;or

m"

Q#

126.

2,m

"R

#20

.8,r

#17

.69.

q"

12,r

"11

,m#

R"

16m

"P

#14

6.5,

m"

Q#

17.5

,p#

22.0

;or

m"

P#

1.5,

m"

Q#

162.

5,p

#1.

010

.p"

29,q

"34

,m#

Q"

111

m"

P#

52.8

,m"

R#

16.2

,r#

10.2

11.I

f m#

P"

89,p

"16

,r"

12m

"Q

#42

.4,m

"R

#48

.6,q

#10

.8

12.I

f m#

Q"

103,

m#

P"

63,p

"13

m"

R#

14,q

#14

.2,r

#3.

5

13.I

f m#

P"

96,m

#R

"82

,r"

35m

"Q

#2,

p#

35.2

,q#

1.2

14.I

f m#

R"

49,m

#Q

"76

,r"

26m

"P

#55

,p#

28.2

,q#

33.4

15.I

f m#

Q"

31,m

#P

"52

,p"

20m

"R

#97

,q#

13.1

,r#

25.2

16.I

f q"

8,m

#Q

"28

,m#

R"

72m

"P

#80

,p#

16.8

,r#

16.2

17.I

f r"

15,p

"21

,m#

P"

128

m"

Q#

17.7

,m"

R#

34.3

,q#

8.1

©G

lenc

oe/M

cGra

w-Hi

ll38

4G

lenc

oe G

eom

etry

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

from

!E

FG

.Rou

nd

an

gle

mea

sure

sto

th

e n

eare

st t

enth

deg

ree

and

sid

e m

easu

res

to t

he

nea

rest

ten

th.

1.If

m#

G"

14,m

#E

"67

,and

e"

14,f

ind

g.3.

7

2.If

e"

12.7

,m#

E"

42,a

nd m

#F

"61

,fin

d f.

16.6

3.If

g"

14,f

"5.

8,an

d m

#G

"83

,fin

d m

#F

.24

.3

4.If

e"

19.1

,m#

G"

34,a

nd m

#E

"56

,fin

d g.

12.9

5.If

f"

9.6,

g"

27.4

,and

m#

G"

43,f

ind

m#

F.

13.8

Sol

ve e

ach

!S

TU

des

crib

ed b

elow

.Rou

nd

mea

sure

s to

th

e n

eare

st t

enth

.

6.m

#T

"85

,s"

4.3,

t"

8.2

m"

S#

31.5

,m"

U#

63.5

,u#

7.4

7.s

"40

,u"

12,m

#S

"37

m"

T#

132.

6,m

"U

#10

.4,t

#48

.9

8.m

#U

"37

,t"

2.3,

m#

T"

17m

"S

#12

6,s

#6.

4,u

#4.

7

9.m

#S

"62

,m#

U"

59,s

"17

.8m

"T

#59

,t#

17.3

,u#

17.3

10.t

"28

.4,u

"21

.7,m

#T

"66

m"

S#

69.7

,m"

U#

44.3

,s#

29.2

11.m

#S

"89

,s"

15.3

,t"

14m

"T

#66

.2,m

"U

#24

.8,u

#6.

4

12.m

#T

"98

,m#

U"

74,u

"9.

6m

"S

#8,

s#

1.4,

t#9.

9

13.t

"11

.8,m

#S

"84

,m#

T"

47m

"U

"49

,s#

16.0

,u#

12.2

14.I

ND

IREC

T M

EASU

REM

ENT

To f

ind

the

dist

ance

fro

m t

he e

dge

of t

he la

ke t

o th

e tr

ee o

n th

e is

land

in t

he la

ke,H

anna

h se

t up

atr

iang

ular

con

figu

rati

on a

s sh

own

in t

he d

iagr

am.T

he d

ista

nce

from

loca

tion

Ato

loca

tion

Bis

85

met

ers.

The

mea

sure

s of

the

an

gles

at

Aan

d B

are

51°

and

83°,

resp

ecti

vely

.Wha

t is

the

dis

tanc

efr

om t

he e

dge

of t

he la

ke a

t B

to t

he t

ree

on t

he is

land

at

C?

abou

t 91.

8 m

A

C

B

Pra

ctic

e (A

vera

ge)

The

Law

of S

ines

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-6

7-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csTh

e La

w o

f Sin

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-6

7-6

©G

lenc

oe/M

cGra

w-Hi

ll38

5G

lenc

oe G

eom

etry

Lesson 7-6

Pre-

Act

ivit

yH

ow a

re t

rian

gles

use

d i

n r

adio

ast

ron

omy?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

6 at

the

top

of

page

377

in y

our

text

book

.

Why

mig

ht s

ever

al a

nten

nas

be b

ette

r th

an o

ne s

ingl

e an

tenn

a w

hen

stud

ying

dis

tant

obj

ects

?Sa

mpl

e an

swer

:Obs

ervi

ng a

n ob

ject

from

onl

y on

e po

sitio

n of

ten

does

not

pro

vide

eno

ugh

info

rmat

ion

to c

alcu

late

thin

gs s

uch

as th

e di

stan

ce fr

om th

eob

serv

er to

the

obje

ct.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.A

ccor

ding

to

the

Law

of

Sine

s,w

hich

of

the

fo

llow

ing

are

corr

ect

stat

emen

ts?

A,F

A.!

sinm

M!"

! sinn

N!"

! sinp

P!B

.!si

n Mm !

"!si

Nnn

!"

!sin P

p!

C.!

coms

M !"

!cos n

N !"

!cops

P !D

.!si

n mM !

%!si

n nN !

"!si

n pP !

E.(

sin

M)2

%(s

in N

)2"

(sin

P)2

F.!si

n pP !

"!si

n mM !

"!si

n nN !

2.St

ate

whe

ther

eac

h of

the

fol

low

ing

stat

emen

ts is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,ex

plai

n w

hy.

a.T

he L

aw o

f Si

nes

appl

ies

to a

ll tr

iang

les.

true

b.T

he P

ytha

gore

an T

heor

em a

pplie

s to

all

tria

ngle

s.Fa

lse;

sam

ple

answ

er:I

ton

ly a

pplie

s to

righ

t tria

ngle

s.c.

If y

ou a

re g

iven

the

leng

th o

f on

e si

de o

f a

tria

ngle

and

the

mea

sure

s of

any

tw

oan

gles

,you

can

use

the

Law

of

Sine

s to

fin

d th

e le

ngth

s of

the

oth

er t

wo

side

s.tru

ed.

If y

ou k

now

the

mea

sure

s of

tw

o an

gles

of

a tr

iang

le,y

ou s

houl

d us

e th

e L

aw o

f Si

nes

to f

ind

the

mea

sure

of

the

thir

d an

gle.

Fals

e;sa

mpl

e an

swer

:You

sho

uld

use

the

Angl

e Su

m T

heor

em.

e.A

fri

end

tells

you

tha

t in

tri

angl

e R

ST

,m#

R"

132,

r"

24 c

enti

met

ers,

and

s"

31ce

ntim

eter

s.C

an y

ou u

se t

he L

aw o

f Si

nes

to s

olve

the

tri

angl

e? E

xpla

in.

No;

sam

ple

answ

er:I

n an

y tri

angl

e,th

e lo

nges

t sid

e is

opp

osite

the

larg

est

angl

e.Be

caus

e a

trian

gle

can

have

onl

y on

e ob

tuse

ang

le,"

Rm

ust b

eth

e la

rges

t ang

le,b

ut s

(r,

so it

is im

poss

ible

to h

ave

a tri

angl

e w

ithth

e gi

ven

mea

sure

s.

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stud

ents

rem

embe

r m

athe

mat

ical

equ

atio

ns a

nd f

orm

ulas

bet

ter

if t

hey

can

stat

eth

em in

wor

ds.S

tate

the

Law

of

Sine

s in

you

r ow

n w

ords

wit

hout

usi

ng v

aria

bles

or

mat

hem

atic

al s

ymbo

ls.

Sam

ple

answ

er:I

n an

y tri

angl

e,th

e ra

tio o

f the

sin

e of

an

angl

e to

the

leng

th o

f the

opp

osite

sid

e is

the

sam

e fo

r all

thre

e an

gles

.

P

MN

p

mn

©G

lenc

oe/M

cGra

w-Hi

ll38

6G

lenc

oe G

eom

etry

Iden

titie

sA

n id

enti

tyis

an

equa

tion

tha

t is

tru

e fo

r al

l val

ues

of t

he

vari

able

for

whi

ch b

oth

side

s ar

e de

fine

d.O

ne w

ay t

o ve

rify

an

iden

tity

is t

o us

e a

righ

t tr

iang

le a

nd t

he d

efin

itio

ns f

ortr

igon

omet

ric

func

tion

s.

Ver

ify

that

(si

n A

)2$

(cos

A)2

"1

is a

n i

den

tity

.

(sin

A)2

%(c

os A

)2"

&!a c! '2%

&!b c! '2

"!a2

% cb2

!"

!c c2 2!"

1

To c

heck

whe

ther

an

equa

tion

may

be a

n id

enti

ty,y

ou c

an t

est

seve

ral v

alue

s.H

owev

er,s

ince

you

can

not

test

all

valu

es,y

ouca

nnot

be

cert

ain

that

the

equ

atio

n is

an

iden

tity

.

Test

sin

2x

"2

sin

xco

s x

to s

ee i

f it

cou

ld b

e an

id

enti

ty.

Try

x"

20.U

se a

cal

cula

tor

to e

valu

ate

each

exp

ress

ion.

sin

2x"

sin

402

sin

xco

s x

"2

(sin

20)

(cos

20)

#0.

643

#2(

0.34

2)(0

.940

)#

0.64

3

Sinc

e th

e le

ft a

nd r

ight

sid

es s

eem

equ

al,t

he e

quat

ion

may

be

an id

enti

ty.

Use

tri

angl

e A

BC

show

n a

bove

.Ver

ify

that

eac

h e

quat

ion

is

an i

den

tity

.

1.!c so ins

AA !"

! tan1

A!

2.!t sa inn

BB!

"! co

1 sB!

!c so ins AA!

"!b c!

)!a c!

"!b a!

"! ta

n1A!

!t sa innBB !

"!b a!

)!b c!

"!c a!

"! co

1 sB

!

3.ta

n B

cos

B"

sin

B4.

1#

(cos

B)2

"(s

in B

)2

tan

B co

s B

"!b a!

*!a c!

"!b c!

"si

n B

1(co

s B)

2"

1'

(!a c! )2

"!c c2 2!

'!a c2 2!

"!c2

c' 2a2

!"

!b c22 !or

(sin

B)2

Try

sev

eral

val

ues

for

x t

o te

st w

het

her

eac

h e

quat

ion

cou

ld b

e an

id

enti

ty.

5.co

s 2x

"(c

os x

)2#

(sin

x)2

6.co

s (9

0#

x)"

sin

x

Yes;

see

stud

ents

’wor

k.Ye

s;se

e st

uden

ts’w

ork.

B

AC

ca

b

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-6

7-6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy

Gu

ide

and I

nte

rven

tion

The

Law

of C

osin

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-7

7-7

©G

lenc

oe/M

cGra

w-Hi

ll38

7G

lenc

oe G

eom

etry

Lesson 7-7

The

Law

of

Co

sin

esA

noth

er r

elat

ions

hip

betw

een

the

side

s an

d an

gles

of

any

tria

ngle

is c

alle

d th

e L

aw o

f C

osin

es.Y

ou c

an u

se t

he L

aw o

f C

osin

es if

you

kno

w t

hree

sid

es o

f a

tria

ngle

or

if y

ou k

now

tw

o si

des

and

the

incl

uded

ang

le o

f a

tria

ngle

.

Let !

ABC

be a

ny tr

iang

le w

ith a

, b, a

nd c

repr

esen

ting

the

mea

sure

s of

the

sides

opp

osite

La

w o

f Cos

ines

the

angl

es w

ith m

easu

res

A, B

, and

C, r

espe

ctive

ly. T

hen

the

follo

wing

equ

atio

ns a

re tr

ue.

a2"

b2%

c2#

2bc

cos

Ab2

"a2

%c2

#2a

cco

s B

c2"

a2%

b2#

2ab

cos

C

In !

AB

C,f

ind

c.

c2"

a2%

b2#

2ab

cos

CLa

w of

Cos

ines

c2"

122

%10

2#

2(12

)(10

)cos

48°

a"

12, b

"10

, m#

C"

48

c"

!12

2%

"10

2#

"2(

12)

"(1

0)co

"s

48°

"Ta

ke th

e sq

uare

root

of e

ach

side.

c#

9.1

Use

a ca

lcula

tor.

In !

AB

C,f

ind

m"

A.

a2"

b2%

c2#

2bc

cos

ALa

w of

Cos

ines

72"

52%

82#

2(5)

(8)

cos

Aa

"7,

b"

5, c

"8

49 "

25 %

64 #

80 c

os A

Mul

tiply.

#40

"#

80 c

os A

Subt

ract

89

from

eac

h sid

e.

!1 2!"

cos

ADi

vide

each

sid

e by

#80

.

cos#

1!1 2!

"A

Use

the

inve

rse

cosin

e.

60°

"A

Use

a ca

lcula

tor.

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

from

!A

BC

.Rou

nd

an

gle

mea

sure

sto

th

e n

eare

st d

egre

e an

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

1.If

b"

14,c

"12

,and

m#

A"

62,f

ind

a.13

.5

2.If

a"

11,b

"10

,and

c"

12,f

ind

m#

B.

51

3.If

a"

24,b

"18

,and

c"

16,f

ind

m#

C.

42

4.If

a"

20,c

"25

,and

m#

B"

82,f

ind

b.29

.8

5.If

b"

18,c

"28

,and

m#

A"

59,f

ind

a.24

.3

6.If

a"

15,b

"19

,and

c"

15,f

ind

m#

C.

51

58

7C

B

A

48#

1210

c

C

BA

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll38

8G

lenc

oe G

eom

etry

Use

th

e La

w o

f C

osi

nes

to

So

lve

Pro

ble

ms

You

can

use

the

Law

of

Cos

ines

toso

lve

som

e pr

oble

ms

invo

lvin

g tr

iang

les.

Let !

ABC

be a

ny tr

iang

le w

ith a

, b, a

nd c

repr

esen

ting

the

mea

sure

s of

the

sides

opp

osite

the

Law

of C

osin

esan

gles

with

mea

sure

s A,

B, a

nd C

, res

pect

ively.

The

n th

e fo

llowi

ng e

quat

ions

are

true

.a2

"b2

%c2

#2b

cco

s A

b2"

a2%

c2#

2ac

cos

Bc2

"a2

%b2

#2a

bco

s C

Ms.

Jon

es w

ants

to

pu

rch

ase

a p

iece

of

lan

d w

ith

th

e sh

ape

show

n.F

ind

th

e p

erim

eter

of

the

pro

per

ty.

Use

the

Law

of

Cos

ines

to

find

the

val

ue o

f a.

a2"

b2%

c2#

2bc

cos

ALa

w of

Cos

ines

a2"

3002

%20

02#

2(30

0)(2

00)

cos

88°

b"

300,

c"

200,

m#

A"

88

a"

!13

0,0

"00

#"

120,

0"

00 c

os"

88°

"Ta

ke th

e sq

uare

root

of e

ach

side.

#35

4.7

Use

a ca

lcula

tor.

Use

the

Law

of

Cos

ines

aga

in t

o fi

nd t

he v

alue

of c

.

c2"

a2%

b2#

2ab

cos

CLa

w of

Cos

ines

c2"

354.

72%

3002

#2(

354.

7)(3

00)

cos

80°

a"

354.

7, b

"30

0, m

#C

"80

c"

!21

5,8

"12

.09

"#

21"

2,82

0"

cos

8"

0°"Ta

ke th

e sq

uare

root

of e

ach

side.

#42

2.9

Use

a ca

lcula

tor.

The

per

imet

er o

f th

e la

nd is

300

%20

0 %

422.

9 %

200

or a

bout

122

3 fe

et.

Dra

w a

fig

ure

or

dia

gram

to

go w

ith

eac

h e

xerc

ise

and

mar

k i

t w

ith

th

e gi

ven

info

rmat

ion

.Th

en s

olve

th

e p

robl

em.R

oun

d a

ngl

e m

easu

res

to t

he

nea

rest

deg

ree

and

sid

e m

easu

res

to t

he

nea

rest

ten

th.

1.A

tri

angu

lar

gard

en h

as d

imen

sion

s 54

fee

t,48

fee

t,an

d 62

fee

t.F

ind

the

angl

es a

t ea

chco

rner

of

the

gard

en.

75°;

48°;

57°

2.A

par

alle

logr

am h

as a

68°

angl

e an

d si

des

8 an

d 12

.Fin

d th

e le

ngth

s of

the

dia

gona

ls.

11.7

;16.

73.

An

airp

lane

is s

ight

ed f

rom

tw

o lo

cati

ons,

and

its

posi

tion

for

ms

an a

cute

tri

angl

e w

ith

them

.The

dis

tanc

e to

the

air

plan

e is

20

mile

s fr

om o

ne lo

cati

on w

ith

an a

ngle

of

elev

atio

n 48

.0°,

and

40 m

iles

from

the

oth

er lo

cati

on w

ith

an a

ngle

of

elev

atio

n of

21.

8°.

How

far

apa

rt a

re t

he t

wo

loca

tion

s?50

.5 m

i4.

A r

ange

r to

wer

at

poin

t A

is d

irec

tly

nort

h of

a r

ange

r to

wer

at

poin

t B

.A fi

re a

t po

int

Cis

obs

erve

d fr

om b

oth

tow

ers.

The

dis

tanc

e fr

om t

he f

ire

to t

ower

Ais

60

mile

s,an

d th

edi

stan

ce f

rom

the

fir

e to

tow

er B

is 5

0 m

iles.

If m

#A

CB

"62

,fin

d th

e di

stan

ce b

etw

een

the

tow

ers.

57.3

mi

200

ft

300

ft

300

ft

88#80

#c

a

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

The

Law

of C

osin

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-7

7-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

The

Law

of C

osin

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-7

7-7

©G

lenc

oe/M

cGra

w-Hi

ll38

9G

lenc

oe G

eom

etry

Lesson 7-7

In !

RS

T,g

iven

th

e fo

llow

ing

mea

sure

s,fi

nd

th

e m

easu

re o

f th

e m

issi

ng

sid

e.

1.r

"5,

s"

8,m

#T

"39

t#5.

2

2.r

"6,

t"

11,m

#S

"87

s#

12.3

3.r

"9,

t"

15,m

#S

"10

3s

#19

.2

4.s

"12

,t"

10,m

#R

"58

r#10

.8

In !

HIJ

,giv

en t

he

len

gth

s of

th

e si

des

,fin

d t

he

mea

sure

of

the

stat

ed a

ngl

e to

th

en

eare

st t

enth

.

5.h

"12

,i"

18,j

"7;

m#

H24

.7

6.h

"15

,i"

16,j

"22

;m#

I46

.7

7.h

"23

,i"

27,j

"29

;m#

J70

.4

8.h

"37

,i"

21,j

"30

;m#

H91

.3

Det

erm

ine

wh

eth

er t

he

Law

of

Sin

esor

th

e L

aw o

f C

osin

essh

ould

be

use

d f

irst

to

solv

e ea

ch t

rian

gle.

Th

en s

olve

eac

h t

rian

gle.

Rou

nd

an

gle

mea

sure

s to

th

e n

eare

std

egre

e an

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

9.10

.

Cosi

nes;

m"

A #

34;

Sine

s;m

"L

#67

;m

"B

#80

;c#

30.7

m"

N#

27;!

#47

.8

11.a

"10

,b"

14,c

"19

12.a

"12

,b"

10,m

#C

"27

Cosi

nes;

m"

A#

31;

Cosi

nes;

m"

A#

97;

m"

B#

46;m

"C

#10

3m

"B

#56

;c#

5.5

Sol

ve e

ach

!R

ST

des

crib

ed b

elow

.Rou

nd

mea

sure

s to

th

e n

eare

st t

enth

.

13.r

"12

,s"

32,t

"34

m"

R#

20.7

,m"

S#

70.2

,m"

T#

89.1

14.r

"30

,s"

25,m

#T

"42

m"

R#

82.2

,m"

S#

55.7

,t#

20.3

15.r

"15

,s"

11,m

#R

"67

m"

S#

42.5

,m"

T#

70.5

,t#

15.4

16.r

"21

,s"

28,t

"30

m"

R#

42.3

,m"

S#

63.8

,m"

T#

74.0

M

LN

!86

#

52

24

B

AC

c

66#

33

19

©G

lenc

oe/M

cGra

w-Hi

ll39

0G

lenc

oe G

eom

etry

In !

JK

L,g

iven

th

e fo

llow

ing

mea

sure

s,fi

nd

th

e m

easu

re o

f th

e m

issi

ng

sid

e.

1.j"

1.3,

k"

10,m

#L

"77

!#

9.8

2.j"

9.6,

!"

1.7,

m#

K"

43k

#8.

43.

j"11

,k"

7,m

#L

"63

!#

10.0

4.k

"4.

7,!

"5.

2,m

#J

"11

2j#

8.2

In !

MN

Q,g

iven

th

e le

ngt

hs

of t

he

sid

es,f

ind

th

e m

easu

re o

f th

e st

ated

an

gle

toth

e n

eare

st t

enth

.

5.m

"17

,n"

23,q

"25

;m#

Q75

.76.

m"

24,n

"28

,q"

34;m

#M

44.2

7.m

"12

.9,n

"18

,q"

20.5

;m#

N60

.28.

m"

23,n

"30

.1,q

"42

;m#

Q10

3.7

Det

erm

ine

wh

eth

er t

he

Law

of

Sin

es o

r th

e L

aw o

f C

osin

es s

hou

ld b

e u

sed

fir

st t

oso

lve

!A

BC

.Th

en s

ole

each

tri

angl

e.R

oun

d a

ngl

e m

easu

res

to t

he

nea

rest

deg

ree

and

sid

e m

easu

re t

o th

e n

eare

st t

enth

.

9.a

"13

,b"

18,c

"19

10.a

"6,

b"

19,m

#C

"38

Cosi

nes;

m"

A#

41;

Cosi

nes;

m"

A#

15;

m"

B#

65;m

"C

#74

m"

B#

127;

c#

14.7

11.a

"17

,b"

22,m

#B

"49

12.a

"15

.5,b

"18

,m#

C"

72

Sine

s;m

"A

#36

;Co

sine

s;m

"A

#48

;m

"C

#95

;c#

29.0

m"

B#

60;c

#19

.8

Sol

ve e

ach

!F

GH

des

crib

ed b

elow

.Rou

nd

mea

sure

s to

th

e n

eare

st t

enth

.

13.m

#F

"54

,f"

12.5

,g"

11m

"G

#45

.4,m

"H

#80

.6,h

#15

.214

.f"

20,g

"23

,m#

H"

47m

"F

#57

.4,m

"G

#75

.6,h

#17

.415

.f"

15.8

,g"

11,h

"14

m"

F#

77.4

,m"

G#

42.8

,m"

H#

59.8

16.f

"36

,h"

30,m

#G

"54

m"

F#

73.1

,m"

H#

52.9

,g#

30.4

17.R

EAL

ESTA

TET

he E

spos

ito

fam

ily p

urch

ased

a t

rian

gula

r pl

ot o

f la

nd o

n w

hich

the

ypl

an t

o bu

ild a

bar

n an

d co

rral

.The

leng

ths

of t

he s

ides

of

the

plot

are

320

fee

t,28

6 fe

et,

and

305

feet

.Wha

t ar

e th

e m

easu

res

of t

he a

ngle

s fo

rmed

on

each

sid

e of

the

pro

pert

y?

65.5

,54.

4,60

.1

Pra

ctic

e (A

vera

ge)

The

Law

of C

osin

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-7

7-7



Rea

din

g t

o L

earn

Math

emati

csTh

e La

w o

f Cos

ines

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-7

7-7

©G

lenc

oe/M

cGra

w-Hi

ll39

1G

lenc

oe G

eom

etry

Lesson 7-7

Pre-

Act

ivit

yH

ow a

re t

rian

gles

use

d i

n b

uil

din

g d

esig

n?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

7 at

the

top

of

page

385

in y

our

text

book

.

Wha

t co

uld

be a

dis

adva

ntag

e of

a t

rian

gula

r ro

om?

Sam

ple

answ

er:

Furn

iture

will

not

fit i

n th

e co

rner

s.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.A

ccor

ding

to

the

Law

of

Cos

ines

,whi

ch

stat

emen

ts a

re c

orre

ct f

or !

DE

F?

B,E,

HA

.d2

"e2

%f2

#ef

cos

DB

.e2

"d

2%

f2#

2df

cos

E

C.d

2"

e2%

f2%

2ef

cos

DD

.f2

"d2

%e2

#2e

fco

s F

E.f

2"

d2%

e2#

2de

cos

FF.

d2

"e2

%f2

G.!

sin d

D !"

!sin e

E !"

!sin f

F !H

.d"

!e2

%f

"2

#2e

"f

cos

"D"

2.E

ach

of t

he f

ollo

win

g de

scri

bes

thre

e gi

ven

part

s of

a t

rian

gle.

In e

ach

case

,ind

icat

ew

heth

er y

ou w

ould

use

the

Law

of

Sine

s or

the

Law

of

Cos

ines

fir

st in

sol

ving

a t

rian

gle

wit

h th

ose

give

n pa

rts.

(In

som

e ca

ses,

only

one

of

the

two

law

s w

ould

be

used

in s

olvi

ngth

e tr

iang

le.)

a.SS

S La

w o

f Cos

ines

b.A

SA L

aw o

f Sin

esc.

AA

S La

w o

f Sin

esd.

SAS

Law

of C

osin

ese.

SSA

Law

of S

ines

3.In

dica

te w

heth

er e

ach

stat

emen

t is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,exp

lain

why

.

a.T

he L

aw o

f C

osin

es a

pplie

s to

rig

ht t

rian

gles

. tru

eb.

The

Pyt

hago

rean

The

orem

app

lies

to a

cute

tri

angl

es.F

alse

;sam

ple

answ

er:

It on

ly a

pplie

s to

righ

t tria

ngle

s.c.

The

Law

of

Cos

ines

is u

sed

to f

ind

the

thir

d si

de o

f a

tria

ngle

whe

n yo

u ar

e gi

ven

the

mea

sure

s of

tw

o si

des

and

the

noni

nclu

ded

angl

e.Fa

lse;

sam

ple

answ

er:I

t is

used

whe

n yo

u ar

e gi

ven

the

mea

sure

s of

two

side

s an

d th

e in

clud

edan

gle.

d.T

he L

aw o

f C

osin

es c

an b

e us

ed t

o so

lve

a tr

iang

le in

whi

ch t

he m

easu

res

of t

he t

hree

side

s ar

e 5

cent

imet

ers,

8 ce

ntim

eter

s,an

d 15

cen

tim

eter

s.Fa

lse;

sam

ple

answ

er:5

$8

+15

,so,

by th

e Tr

iang

le In

equa

lity

Theo

rem

,no

such

trian

gle

exis

ts.

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al f

orm

ula

is t

o re

late

it t

o on

e yo

u al

read

ykn

ow.T

he L

aw o

f C

osin

es lo

oks

som

ewha

t lik

e th

e P

ytha

gore

an T

heor

em.B

oth

form

ulas

mus

t be

tru

e fo

r a

righ

t tr

iang

le.H

ow c

an t

hat

be?

cos

90 "

0,so

in a

righ

ttri

angl

e,w

here

the

incl

uded

ang

le is

the

right

ang

le,t

he L

aw o

f Cos

ines

beco

mes

the

Pyth

agor

ean

Theo

rem

.

D

dE

e F

f

©G

lenc

oe/M

cGra

w-Hi

ll39

2G

lenc

oe G

eom

etry

Sphe

rical

Tria

ngle

sSp

heri

cal t

rigo

nom

etry

is a

n ex

tens

ion

of p

lane

tri

gono

met

ry.

Fig

ures

are

dra

wn

on t

he s

urfa

ce o

f a

sphe

re.

Arc

s of

gre

at

circ

les

corr

espo

nd t

o lin

e se

gmen

ts in

the

pla

ne.

The

arc

s of

th

ree

grea

t ci

rcle

s in

ters

ecti

ng o

n a

sphe

re f

orm

a s

pher

ical

tr

iang

le.

Ang

les

have

the

sam

e m

easu

re a

s th

e ta

ngen

t lin

es

draw

n to

eac

h gr

eat

circ

le a

t th

e ve

rtex

.Si

nce

the

side

s ar

e ar

cs,t

hey

too

can

be m

easu

red

in d

egre

es.

Sol

ve t

he

sph

eric

al t

rian

gle

give

n a

"72

#,b

"10

5#,a

nd

c"

61#.

Use

the

Law

of

Cos

ines

.

0.30

90"

(–0.

2588

)(0.

4848

)%(0

.965

9)(0

.874

6) c

os A

cos

A"

0.51

43A

"59

°

#0.

2588

" (0

.309

0)(0

.484

8)%

(0.9

511)

(0.8

746)

cos

Bco

s B

"#

0.49

12B

"11

9°

0.48

48"

(0.3

090)

(–0.

2588

)%(0

.951

1)(0

.965

9) c

os C

cos

C"

0.61

48C

"52

°

Che

ck b

y us

ing

the

Law

of

Sine

s.

!s si in n7 52 9° °!

"!s si in n

1 10 15 9° °!

"!s si in n

6 51 2° °!

"1.

1

Sol

ve e

ach

sp

her

ical

tri

angl

e.

1.a

"56

°,b

"53

°,c

"94

°2.

a"

110°

,b"

33°,

c"

97°

A"

41#,

B"

39#,

C"

128#

A"

116#

,B"

31#,

C"

71#

3.a

"76

°,b

"11

0°,C

"49

°4.

b"

94°,

c"

55°,

A"

48°

A"

59#,

B"

124#

,c"

59#

a"

60#,

B"

121#

,C"

45#

A

C

B

c

ba

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

7-7

7-7

The

sum

of t

he s

ides

of a

sph

erica

l tria

ngle

is le

ss th

an 3

60°.

The

sum

of t

he a

ngle

s is

grea

ter t

han

180°

and

less

than

540

°.Th

e La

w of

Sin

es fo

r sph

erica

l tria

ngle

s is

as fo

llows

.

! ss ii nnAa !

"! ss ii nn

Bb !"

! ss ii nn Cc!

Ther

e is

also

a L

aw o

f Cos

ines

for s

pher

ical t

riang

les.

cos

a"

cos

bco

s c

%sin

bsin

cco

s A

cos

b"

cos

aco

s c

%sin

asin

cco

s B

cos

c"

cos

aco

s b

%sin

asin

bco

s C

Exam

ple

Exam

ple


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