lect notes - geometry

8/16/2019 Lect Notes - Geometry

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GEOMETRY

PRE – CALCULUS MODULE

CHAPTER 2 MAP 2163

MAP2163/pre-calculus module/KING

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COURSE CONTENTS

1.

Angle properties

2.

Circle and its properties

3. Triangle properties

4.

Pythagoras theorem

5. Similar triangles

6.

Areas of plane figures

7. Areas and volumes of regular solids,and frusta of pyramids, cones andspheres


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LESSON OBJECTIVES

You should be able to……

1. Convert angles into degree and radians.

2. Add and subtract angles in degrees, minutesand seconds.

3.

Identify types of angles, and use itsproperties to calculate unknown angles.

4. Name the major parts of a circle, and applyformula to calculate the parts.

5.

Use the properties of angles in a circle tocalculate unknown angles.

6. Use the properties of similar and congruenttriangles to calculate unknown sides.

7.

Apply Pythagoras theorem to calculateunknown sides in a right-angled triangle.

8. Find the areas of plane figures.


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9. Apply formula to find areas and volumes ofregular solids, and frusta of pyramids, conesand spheres.

10. Solve practical problems involvingmensuration.

ANGLE PROPERTIES

An angle is formed by the intersection of two straight lines. It can be measured indegrees or radians.

If the line OB lies along OA, then it is rotated until it lies along OA again, it is said to

make one revolution, or rotated at 360 .

061

061

360revolution1

Types of angles

i) Acute angle:

An angle between 0 to 90


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ii) Right angle:

An angle at 90

iii) Obtuse angle:

An angle between

90

to

180

iv) Reflex angle:

An angle between 180 to 360

A straight line which crosses two parallel lines is called a transversal. Theproperties of angles are shown in figures below:

i) Corresponding angle:

b a


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ii) Alternate angle:

b a

iii) Opposite angle:

b a

iv)

Interior angle:

180b a

When two straight lines intersect, the opposite angles are equal:


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d b

c a

When 2 angles make up 90 , they are called complementary angles:

When 2 angles make up

180 , they are called supplementary angles:

Radian measure

Consider an arc of length 1 unit, and radius 1 unit. The angle subtended at thecentre of the circle by the arc is 1 rad. The circumference of a circle is given by

r C 2 . Substituting 1r gives us 212 C .Therefore, there are 2 radiansat the centre of the circle. In short, 2 radians is equivalent to 360 .


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Convert angles into degrees and radians

To convert degrees into radians, we multiply by

180

.

To convert radians into degrees, we multiply by

180

.

Example:

1.

Express 270° in radians.

We know that

rad2

3270

270rad3602270

rad360

21

rad2360


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2. Express 3

4radians in degrees.

We know that

240rad3

4

rad3

4

2

360rad

3

4

2

360rad1

360rad2

Add and subtract angles in degrees, minutes

and seconds

Example:

1. Add the following angles 116426 and 158336

264862

158336

116426

25263

63162

5206148

20626


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2. Subtract 811115 from 723324

902209

721115

813324

Since 81 cannot minus 72 , we borrow 1 from the minutes on the left.

15129

91524

1211133

15720681


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EXERCISE 2.1


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CIRCLE PROPERTIES

Circle properties summarised in the figure below:


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Formula regarding circle1.

Circumference (Perimeter) = r 2

2. Area 2r

3. Arc r 2360

, is measured in degrees.

Arc r , is measured in radians.

4.

Sector 2360 r

, is measured in degrees.

Sector 2

2

1r , is measured in radians.

Example:

1. Calculate the length of the circumference of a circle of a radius 6.50 cm.

Circumference r 2

cm 84.40

50.62

2. Find the length of arc of a circle of radius 4.23 cm when the angle subtendedat the centre is 1.46 radians.

Arc r , is measured in radians.

cm

rad

176.6

46.123.4


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Angles in a circle Angles subtended from the same arc, ondifferent vertices on the circumferenceare equal to each other.

An angle subtended from an arc to thecentre is twice the angle subtended fromthe same arc to a point on thecircumference.

Angles formed by drawing lines from theends of the diameter of a circle to itscircumference form a right angle.

EXERCISE 2.2


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TRIANGLE PROPERTIES

A triangle is a 3-sided polygon, with 3 angles that sum up to 180 .

180c b a

Types of triangle

i) Equilateral

BC AC AB

ii) Isosceles

BC AC


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iii) Scalene

BC AC AB

c b a

Congruent:Two triangles are said to be congruent if they are equal in every aspects, the anglesand the sides.

EF DF DE BC AC AB

Similar:Two triangles are said to be similar if they have the same angles respectively butdiffer in the length of sides, respectively.

DF DE BC AC


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Outer angle:In the figure below, the exterior angle is equal to the sum of the opposite interior

angles.

b a

Example:

1. Find the angles x and y .

80y (alternate angle), and

1013040180x

( 40 is the angle subtended from the same arc as the angle 80 . 130 is the

angle y 50 . The triangle at centre O is an isosceles triangle, since the

sides are the circle’s radius. So

50280180 )


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2. These two triangles are similar. Find the value t .

PQ

ST

PR

SU

cm t

t

t

8.4

610

8

610

8


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EXERCISE 2.3


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The Theorem of Pythagoras

In any right-angled triangle, the square on the hypotenuse is equal to the sum ofsquares on the other two sides.

In the right-angled triangle ABC above, b is the hypotenuse. (Hypotenuse is always

on the opposite side of the right angle).

Therefore, 222 c a b

It is good to know some of the basic arrangement of Pythagoras theorem, i.e:

Example:

In the following figure, AB is tangent to the circle at B. If the circle has a radius 40mm, and AB = 150 mm, find the length of AO.


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Solution:

The tangent that touches point B on the circle makes 90 ABO . Hence, we can

use Pythagoras theorem to calculate AO.

mm AO

AO

AO

AO

24.155

24100

225001600

15040

2

222


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EXERCISE 2.4


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EXERCISE 2.5


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AREAS OF PLANE FIGURES

i) Square

Area, 2 x A

ii) Rectangle

Area, wl A

iii) Parallelogram

Area, hb A


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iv) Trapezium

Area, hba A 2

1

v) Triangle

Area, hb A2

1

vi) Circle

Area, 2r A


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EXERCISE 2.6


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VOLUMES OF REGULAR SOLIDS

i) Rectangular solid

hwl V

ii) Prism

h BV

*B is the area of the base surface

iii) Cylinder

rhr A 22 2

hr V 2


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iv) Pyramid

h BV 3

1

v) Cone

rsr A 2 , ( 22 hr s )

hr V 2

3

1

vi) Sphere

24 r A

3

3

4r V


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EXERCISE 2.7


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AREAS AND VOLUMES

OF FRUSTUMS A frustum is a portion of a solid, normally a cone or a pyramid, which lies betweentwo parallel planes cutting it.

The easy way to calculate the volume of a frustum is by subtracting the volume ofthe cut portion from the volume of the original portion.

The volume of frustum ABCD = [volume of cone AEB] – [volume of cone CED]


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Frustum of a cone

r = radius of the upper baseR = radius of the lower baseh = height of the frustums = slant height of the frustum

Example:

The radii of the faces of a frustum of a cone are 3 cm and 4 cm, and its height is 5cm. Find its volume.

Solution: Always sketch your frustum first. Put the values right before you calculate.

To find x , we use the proportion of the similar shape:


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cm x

x

15

1

5

3

The height of the original cone is cm20155

Volume of original cone is 321 103.3352043

1cmV

Volume of the small cut cone is 322 372.1411533

1cmV

Volume of frustum is 321 7.193 cmV V V

Frustum of pyramid

* Similar to calculating volume of conical frustum.

Assignment:

A hole is to be dug in the form of a frustum of a pyramid. The top is to be a squareof side 6.40 m, and the bottom a square of 3.60 m. If the depth of the hole is to be4.00 m, calculate the volume of the earth to be removed.If the hole is now filled with concrete to a depth of 2.00 m, find the amount ofconcrete required.

[ Answer: 102.6 m 3 , and 43.1 m 3 ]

lect notes - geometry

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