Chapter 10 Geometry 2010 Pearson Education, Inc. All rights reserved

Chapter 10

Geometry

© 2010 Pearson Education, Inc.All rights reserved.

10.1 Basic Geometric TermsObjectives

Slide 8.1- 2

1. Identify and name lines, line segments, and rays. 2. Identify parallel and intersecting lines. 3. Identify and name angles. 4. Classify angles as right, acute, straight, or

obtuse. 5. Identify perpendicular lines.

Copyright © 2010 Pearson Education, Inc. All rights reserved.

Slide 8.1- 3

Geometry starts with the idea of a point. A point is a location in space. It has no length or width. A point is represented by a dot and is named by writing a capital letter next to the dot.

R

Point R


Slide 8.1- 4

A line is a straight row of points that goes on forever in both directions.A line is named using the letters of any two points on the line.

A piece of line that has two endpoints is called a line segment.

AB�

AB


Slide 8.1- 5

A ray is a part of a line that has only one endpoint and goes on forever in one direction.

The endpoint is always written first when naming a ray.

AB


Identify each figure below as a line, line segment, or ray, and name it using the appropriate symbol.a. b. c.

ParallelExample 1 Identifying and Naming Lines, Rays,

and Line Segments

Slide 8.1- 6

line segment BC ray BA

line DE�


Slide 8.1- 7

A plane is an infinitely large flat surface. A floor or a wall is part of a plane.

Lines that are in the same plane, but that never intersect (never cross), are called parallel lines, while lines that cross are called intersecting lines.


Label each pair of lines as appearing to be parallel or as intersecting.a. b. c.

ParallelExample 2 Identifying Parallel and Intersecting

Lines

Slide 8.1- 8

appear parallel intersecting appear parallel


Slide 8.1- 9

An angle is made up of two rays that start at a common endpoint. This common endpoint is called the vertex.


Slide 8.1- 10 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Name the highlighted angle (pink) in three different ways.

ParallelExample 3 Identifying and Naming an Angle

Slide 8.1- 11

EGF

E

F

G

H

J

2 3

4

FGE 2


Slide 8.1- 12

Angles can be measured in degrees. The symbol for degrees is a small raised circle .

An angle of 180 is called a straight angle.

An angle of 90 is called a right angle.


Slide 8.1- 13

Some other terms used to describe angles are shown below.

Acute angles measure less than 90

Obtuse angles measure more than 90 but less than 180.


Slide 8.1- 14


Label each angle as acute, right, obtuse, or straight.a. b.

c. d.

ParallelExample 4 Classifying Angles

Slide 8.1- 15

straight angleacute angle

obtuse angleright angle


Slide 8.1- 16

Two lines are called perpendicular lines if they intersect to form a right angle.

and CB ST� � � � � � � � � � � � � � �

are perpendicular lines because they intersect at right angles.


Which pairs of lines are perpendicular?a. b.

ParallelExample 5 Identifying Perpendicular Lines

Slide 8.1- 17

Perpendicular Intersecting but not perpendicular

A

BC

DE

A

B

C

D


10.2 Rectangles and Squares

Objectives

Slide 8.3- 18

1. Find the perimeter and area of a rectangle.

2. Find the perimeter and area of a square.

3. Find the perimeter and area of a composite figure.


A rectangle has four sides that meet to form 90° angles. Each set of opposite sides is parallel and congruent (has the same length).

5 cm

9 cm

5 cm

9 cm

In a rectangle, if one right angle is shown, the other three are also right angles.

90°angles

Each longer side of a rectangle is called the length (l) and each shorter side is called the width (w).



ParallelExample 1 Finding the Perimeter of a Rectangle

Slide 8.3- 21

Find the perimeter of each rectangle.

a.6 m

16 m

6 m

16 m

P = 2 • l + 2 • wP = 2 • 16 m + 2 • 6 mP = 32 m + 12 mP = 44 m

The perimeter of the rectangle is 44 m. Copyright © 2010 Pearson Education, Inc. All rights reserved.

ParallelExample 1continued

Finding the Perimeter of a Rectangle

Slide 8.3- 22


b. A rectangle 7.8 ft by 12.3 ftP = 2 • l + 2 • w

Either method will give you the same result.

P = 2 • 12.3 ft + 2 • 7.8 ftP = 24.6 ft + 15.6 ftP = 40.2 ft

Or, you can add up the lengths of the four sides.P = 12.3 ft + 12.3 ft + 7.8 ft + 7.8 ftP = 40.2 ft


Slide 8.3- 23

The perimeter of a rectangle is the distance around the outside edges.

The area of a rectangle is the amount of surface inside the rectangle.

8 m

5 m

1 m

1 m

We have five rows of eight square meters for a total of 40 square meters.

1 square meter or (m)2



Squares of many sizes can be used to measure area. For smaller areas, you might use the ones shown below.

Slide 8.3- 25

1 square inch(1 in.2)

1 in.

1 in.

1 squarecentimeter

(1 cm2)

1 cm1 cm

1 squaremillimeter(1 mm2)

1 mm1 mm

(Approximate-size drawings)

Other sizes of squares that are often used to measure area:

1 square meter (1 m2) 1 square foot (1 ft2)1 square kilometer (1 km2) 1 square yard (1 yd2)

1 square mile (1 mi2) Copyright © 2010 Pearson Education, Inc. All rights reserved.

ParallelExample 2 Finding the Area of a Rectangle

Slide 8.3- 26

Find the area of each rectangle.

a.

7 yd

15 yd

A = l • wA = 15 yd • 7 ydA = 105 yd2



Finding the Area of a Rectangle

Slide 8.3- 27


b.

A = l • wA = 18 cm • 3 cmA = 54 cm2

18 cm

3 cm



ParallelExample 3 Finding the Perimeter and Area of a

Square

Slide 8.3- 29

a. Find the perimeter of a square where each side measures 7 m.

Use the formula.

P = 4 • sP = 4 • 7 mP = 28 m

Or add up the four sides.

P = 7 m + 7 m + 7 m + 7 mP = 28 m

Same answer



Finding the Perimeter and Area of a Square

Slide 8.3- 30

b. Find the area of a square where each side measures 7 m.

A = s • sA = 7 m • 7 m

A = s2

A = 49 m2 Square units for area.



Composite Figure

Slide 8.3- 31

a. The floor of a room has the shape shown.

6 ft

6 ft

30 ft

21 ft

24 ft15 ft

Suppose you want to put new wallpaper border along the top of the walls. How much material do you need?

Find the perimeter of the room by adding up the length of the sides.

P = 30 ft + 21 ft + 24 ft + 15 ft + 6 ft + 6 ft = 102 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.


Finding the Perimeter and Area of a Composite Figure

Slide 8.3- 32

b. The carpet you like cost $24.25 per square yard. How much will it cost to carpet the room?

First change the measurements from feet to yards.

2 yd

2 yd

10 yd

7 yd

8 yd

5 yd

6 ft

6 ft

30 ft

21 ft

24 ft

15 ft


Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley



Slide 8.3- 33

b. Next break the room into two pieces. Use just the measurements for the length and width of each piece.

2 yd

2 yd

7 yd

8 yd

Area of rectangle = l • wA = 8 yd • 7 ydA = 56 yd2

Area of square = s2

A = s • sA = 2 yd • 2 ydA = 4 yd2

Total area = 56 yd2 + 4 yd2 = 60 yd2



Slide 8.3- 34

b. Finally, multiply to find the cost of the carpet.

2 yd

2 yd

7 yd

8 yd

• $24.251 yd2

60 yd2

1= $1455

It will cost $1455 to carpet the room.


10.1 (cont.) Angles and Their Relationships

Objectives

Slide 8.2- 35

1. Identify complementary angles and supplementary angles and find the measureof a complement or supplement of an angle.

2. Identify congruent angles and vertical angles and use this knowledge to find the measures of angles.


Slide 8.2- 36

Two angles are called complementary angles if the sum of their measures is 90.

If two angles are complementary, each angle is the complement of the other.


Identify each pair of complementary angles.

ParallelExample 1 Identifying Complementary Angles

Slide 8.2- 37

1

2

3

4

15

1575

75

1 and 2

75 15 90

1 and 4

75 15 90

2 and 3

75 15 90

3 and 4

75 15 90


Find the complement of each angle.

a. 32

b. 65

ParallelExample 2 Finding the Complement of Angles

Slide 8.2- 38

32 5890

65 2590


Slide 8.2- 39

Two angles are called supplementary angles if the sum of their measures is 180.

If two angles are supplementary, each angle is the supplement of the other.


Identify each pair of supplementary angles.

ParallelExample 3 Identifying Supplementary Angles

Slide 8.2- 40

215165

1

3

165

4 15

1 and 2

165 15 180

1 and 4

165 15 180

2 and 3

165 15 180

3 and 4

165 15 180


Find the supplement of each angle.

a. 84

b. 135

ParallelExample 4 Finding the Supplement of Angles

Slide 8.2- 41

84 96180

135 45180


Slide 8.2- 42

Two angles are called congruent angles if they measure the same number of degrees.


Identify the angles that are congruent.

ParallelExample 5 Identifying Congruent Angles

Slide 8.2- 43

JKL NKM 95

858595

J

K

L M

N

JKN LKM



Identify the vertical angles in the figure.

ParallelExample 6 Identifying Vertical Angles

Slide 8.2- 45

A

BC

DE

ABC and EBD

CBD and ABE


In the figure below, find the measure of each unlabeled angle.

ParallelExample 7 Finding the Measure of Vertical

Angles

Slide 8.2- 46

AGF CDG 116

EGD AGB

36 116

36

180 (116 36 )FGE

28

28

28


10.2/10.3 Rectangles and Squares

Objectives

Slide 8.3- 47

1. Find the perimeter and area of a rectangle.

2. Find the perimeter and area of a square.

3. Find the perimeter and area of a composite figure.


A rectangle has four sides that meet to form 90° angles. Each set of opposite sides is parallel and congruent (has the same length).

5 cm

9 cm

5 cm

9 cm

In a rectangle, if one right angle is shown, the other three are also right angles.

90°angles

Each longer side of a rectangle is called the length (l) and each shorter side is called the width (w).



ParallelExample 1 Finding the Perimeter of a Rectangle

Slide 8.3- 50


a.6 m

16 m

6 m

16 m

P = 2 • l + 2 • wP = 2 • 16 m + 2 • 6 mP = 32 m + 12 mP = 44 m

The perimeter of the rectangle is 44 m. Copyright © 2010 Pearson Education, Inc. All rights reserved.


Finding the Perimeter of a Rectangle

Slide 8.3- 51


b. A rectangle 7.8 ft by 12.3 ftP = 2 • l + 2 • w

Either method will give you the same result.

P = 2 • 12.3 ft + 2 • 7.8 ftP = 24.6 ft + 15.6 ftP = 40.2 ft

Or, you can add up the lengths of the four sides.P = 12.3 ft + 12.3 ft + 7.8 ft + 7.8 ftP = 40.2 ft


Slide 8.3- 52

The perimeter of a rectangle is the distance around the outside edges.

The area of a rectangle is the amount of surface inside the rectangle.

8 m

5 m

1 m

1 m

We have five rows of eight square meters for a total of 40 square meters.

1 square meter or (m)2



Squares of many sizes can be used to measure area. For smaller areas, you might use the ones shown below.

Slide 8.3- 54

1 square inch(1 in.2)

1 in.

1 in.

1 squarecentimeter

(1 cm2)

1 cm1 cm

1 squaremillimeter(1 mm2)

1 mm1 mm

(Approximate-size drawings)

Other sizes of squares that are often used to measure area:

1 square meter (1 m2) 1 square foot (1 ft2)1 square kilometer (1 km2) 1 square yard (1 yd2)

1 square mile (1 mi2) Copyright © 2010 Pearson Education, Inc. All rights reserved.

ParallelExample 2 Finding the Area of a Rectangle

Slide 8.3- 55


a.

7 yd

15 yd

A = l • wA = 15 yd • 7 ydA = 105 yd2



Finding the Area of a Rectangle

Slide 8.3- 56


b.

A = l • wA = 18 cm • 3 cmA = 54 cm2

18 cm

3 cm




Square

Slide 8.3- 58

a. Find the perimeter of a square where each side measures 7 m.

Use the formula.

P = 4 • sP = 4 • 7 mP = 28 m

Or add up the four sides.

P = 7 m + 7 m + 7 m + 7 mP = 28 m

Same answer



Finding the Perimeter and Area of a Square

Slide 8.3- 59

b. Find the area of a square where each side measures 7 m.

A = s • sA = 7 m • 7 m

A = s2

A = 49 m2 Square units for area.



Composite Figure

Slide 8.3- 60

a. The floor of a room has the shape shown.

6 ft

6 ft

30 ft

21 ft

24 ft15 ft

Suppose you want to put new wallpaper border along the top of the walls. How much material do you need?

Find the perimeter of the room by adding up the length of the sides.

P = 30 ft + 21 ft + 24 ft + 15 ft + 6 ft + 6 ft = 102 ft Copyright © 2010 Pearson Education, Inc. All rights reserved.



Slide 8.3- 61

b. The carpet you like cost $24.25 per square yard. How much will it cost to carpet the room?

First change the measurements from feet to yards.

2 yd

2 yd

10 yd

7 yd

8 yd

5 yd

6 ft

6 ft

30 ft

21 ft

24 ft

15 ft


Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley



Slide 8.3- 62

b. Next break the room into two pieces. Use just the measurements for the length and width of each piece.

2 yd

2 yd

7 yd

8 yd

Area of rectangle = l • wA = 8 yd • 7 ydA = 56 yd2

Area of square = s2

A = s • sA = 2 yd • 2 ydA = 4 yd2

Total area = 56 yd2 + 4 yd2 = 60 yd2



Slide 8.3- 63

b. Finally, multiply to find the cost of the carpet.

2 yd

2 yd

7 yd

8 yd

• $24.251 yd2

60 yd2

1= $1455

It will cost $1455 to carpet the room.


10.2/10.3 Parallelograms and Trapezoids

Objectives

Slide 8.4- 64

1. Find the perimeter and area of a parallelogram.

2. Find the perimeter and area of a trapezoid.


A parallelogram is a four-sided figure with opposite sides parallel, such as the ones below. Notice that the opposite sides have the same length.


ParallelExample 1 Finding the Perimeter of a

Parallelogram

Slide 8.4- 66

Find the perimeter of a the parallelogram.

P = 15 cm + 9 cm + 15 cm + 9 cm

15 cm

15 cm

9 cm9 cm

= 48 cm



ParallelExample 2 Finding the Area of a Parallelogram

Slide 8.4- 68

Find the area of the parallelogram.

The base is 10 m and the height is 3 m. Use the formula to solve.

10 m

10 m

4 m4 m 3 m

A = b ∙ hA = 10 m ∙ 3 mA = 30 m2


A trapezoid is a four-sided figure with exactly one pair of parallel sides, such as the figures shown below. Unlike the parallelogram, opposite sides of a trapezoid might not have the same length.


ParallelExample 3 Finding the Perimeter of a Trapezoid

Slide 8.4- 70

Find the perimeter of a the trapezoid.

P = 10 m + 13 m + 10 m + 7 m = 40 m

7 m

13 m

8 m10 m 10 m

Notice the height (8 m) is not part of the perimeter, because the height is not one of the outside edges of the shape.



ParallelExample 4 Finding the Area of a Trapezoid

Slide 8.4- 72

Find the area of this trapezoid. The short and long bases are the parallel sides.

7 m

13 m

8 m10 m 10 m

1 ( )2

h BA b

1 (7 )2

m8 m 13 mA

8 m1 (20 m)2

A 1

10

280 mA

Note: You can also use 0.5, the decimal equivalent for ½ in the formula.


ParallelExample 6 Applying Knowledge of Area

Slide 8.4- 73

Suppose the figure in Example 4 represents the floor plan of a hospital lobby. What is the cost to tile the area if tile costs $16.75 per square meter?

The floor area is 80 m2. To find the cost to tile the floor, multiply the number of square meters times the cost of the tile per square meter.

2

2

80 m $16.75cost1 1 m

cost $1340The cost of tile for the lobby is $1340.


10.4 TrianglesObjectives

Slide 8.5- 74

1. Find the perimeter of a triangle.

2. Find the area of a triangle.

3. Given the measures of two angles in a triangle,find the measure of the third angle.


Slide 8.5- 75

A triangle is a figure with exactly three sides.

To find the perimeter of a triangle, add the lengths of the three sides.


Find the perimeter of the triangle.

P = 12 ft + 16 ft + 20 ft = 48 ft

ParallelExample 1 Finding the Perimeter of a Triangle

Slide 8.5- 76

12 ft

16 ft

20 ft


Slide 8.5- 77

The height of a triangle is the distance from one vertex of the triangle to the opposite side (base).

The height line must be perpendicular to the base; that is, it must form a right angle with the base.


Slide 8.5- 78


Find the area of each triangle.

a.

ParallelExample 2 Find the Area of a Triangle

Slide 8.5- 79

12

A b h

52 ft 14 ft12

A

12

A 1

5226

14 ft ft

2364 ftA



b.

ParallelExample 2 continued

Find the Area of a Triangle

Slide 8.5- 80

0.5 bA h

0.5 34.2 14.6A

2249.66 cmA

45.5 cm

34.2 cm

14.6 cm



c.


Find the Area of a Triangle

Slide 8.5- 81

0.5 bA h

0.5 12.75 8.5A

54.1875A

182

3124


Find the area of the shaded part of the figure.

ParallelExample 3 Using the Concept of Area

Slide 8.5- 82

A l w 62 cm 46 cmA

22852 cmA

The entire figure is a rectangle. Find the area.

The unshaded part is a triangle. Find the area of the triangle.

1 62 cm 34 cm2

A

12

A 1

6231

34

21054 cmA

Subtract to find the area of the shaded part.

2 22852 cm 1054 cmA 21798 cmA


Slide 8.5- 83


Find the number of degrees in Angle C.

ParallelExample 5 Finding an Angle Measurement in

Triangles

Slide 8.5- 84

Step 1 Add the two angle measurements you are given.

44 29

731 108 70

107C

73

Step 2 Subtract the sum from 180.


Find the number of degrees in Angle D.

ParallelExample 5

Slide 8.5- 85

Step 1 90 37 127

127 53180

53D

Step 2

is a right angle; it equals 90E

Finding an Angle Measurement in Triangles


10.4 (cont.) Pythagorean TheoremObjectives

Slide 8.8- 86

1. Find square roots using the square root key on a calculator.

2. Find the unknown length in a right triangle.

3. Solve application problems involving right triangles.


A number that has a whole number as its square root is called a perfect square.

The first few perfect squares are listed below.


ParallelExample 1 Find the Square Root of Numbers

Slide 8.8- 88

Use a calculator to find each square root. Round answers to the nearest thousandth.

a. 46

The calculator shows 6.782329983; round to 6.782

b. 136The calculator shows 11.66190379; round to 11.662

c. 260The calculator shows 16.1245155; round to 16.125


Slide 8.8- 89

One place you will use square roots is when working with the Pythagorean Theorem. This theorem applies only to right triangles. Recall that a right triangle is a triangle that has one 90° angle. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called legs.




ParallelExample 2 Find the Unknown Length in Right

Triangles

Slide 8.8- 92

Find the unknown length in each right triangle. Round answers to the nearest tenth if necessary.

a.The unknown length is the side opposite the right angle. Use the formula for finding the hypotenuse.

8 cm

15 cm

2 2hypotenuse = leg leg

2 2hypotenuse = 8 15

= 64 225

= 289 = 17The length is 17 cm. long



Find the Unknown Length in Right Triangles

Slide 8.8- 93


b. Use the formula for finding the leg.

15 ft 2 2leg = hypotenuse leg

= 1600 225 = 1375 37.1

40 ft

2 2leg = 40 15

The length is approximately 37.1 ft long. Copyright © 2010 Pearson Education, Inc. All rights reserved.

ParallelExample 3 Using the Pythagorean Theorem

Slide 8.8- 94

An electrical pole is shown below. Find the length of the guy wire. Round your answer to the nearest tenth of a foot if necessary.



= 1225 3600

= 4825 69.5The length of the guy wire is approximately 69.5 ft.

35ft

60 ft


10.5 Circles

Objectives

Slide 8.6- 95

1. Find the radius and diameter of a circle.

2. Find the circumference of a circle.

3. Find the area of a circle.

4. Become familiar with Latin and Greek prefixesused in math terminology.


Slide 8.6- 96

Suppose you start with one dot on a piece of paper. Then place many dots that are each 3 cm away from the first dot. If we place enough dots (points) we’ll end up with a circle. The 3cm is the radius of the circle. The distance across is the diameter.

Each line below is 3 cm long.3 cm diameter

radius

center

r r

d


Slide 8.6- 97

r r

d


Slide 8.6- 98

r r

d


ParallelExample 1 Finding the Diameter and Radius of a

Circle

Slide 8.6- 99

Find the unknown length of the diameter or radius in each circle.

a.

r = 12 in.d = ?

Because the radius is 12 in., the diameter is twice as long.

d = 2 • rd = 2 • 12 in.d = 24 in.



Finding the Diameter and Radius of a Circle

Slide 8.6- 100

Find the unknown length of the diameter or radius in each circle.

b.

r = ?

d = 7 m

The radius is half the diameter.

r = d2

r = 7 m2

r = 3.5 m or 3 m12


Slide 8.6- 101

The perimeter of a circle is called its circumference. Circumference is the distance around the edge of a circle.


Slide 8.6- 102

Dividing the circumference of any circle by its diameteralways gives an answer close to 3.14.

This means that going around the edge of any circle is a little more than 3 times as far as going straight across the circle.

3.14159265359

This ratio of circumference to diameter is called .



ParallelExample 2 Finding the Circumference of Circles

Slide 8.6- 104

Find the circumference of each circle. Use 3.14 as the approximate value for . Round answers to the nearest tenth.

a.

24 m

The diameter is 24 m, so use the formula with d in it.

C = • d

C = 3.14 • 24 m

C ≈ 75.4 m Rounded


ParallelExample 2 Finding the Circumference of Circles

Slide 8.6- 105

Find the circumference of each circle. Use 3.14 as the approximate value for . Round answers to the nearest tenth.

b.

6.5 cm

In this example, the radius is labeled,so it is easier to use the formula withr in it.

C = 2 • • r

C = 2 • 3.14 • 6.5 cm

C ≈ 40.8 cm Rounded


Slide 8.6- 106

Finding the Area of a CircleC = 2 • π • r

C = 2 • π • r

Unfold each circle. Now put them together.

2 • π • r

2 • π • r

2 • π • r

2 • π • r

r r


Slide 8.6- 107

The figure is approximately a parallelogram.

Area = b • h

Area = 2 • • r • r

2 • • r

r

Area = 22 r

Note: This is the area for 2 circles. The area for one circle is found by using 2.A r



ParallelExample 3 Finding the Area of Circles

Slide 8.6- 109

Find the area of each circle. Use 3.14 for . Round answers to the nearest tenth.

a.

A circle with a radius of 14.2 cm.

Rounded; square units for area

A = • r • rA ≈ 3.14 • 14.2 cm • 14.2 cm

A ≈ 633.1 cm2



Finding the Area of Circles

Slide 8.6- 110

Find the area of each circle. Use 3.14 for . Round answers to the nearest tenth.

b.

Now find the area.

24 ftFirst find the radius.

r = d2

r = = 12 ft24 ft2

A ≈ 3.14 • 12 ft • 12 ft

A ≈ 452.2 ft2


ParallelExample 4 Finding the Area of a Semicircle

Slide 8.6- 111

Find the area of the semicircle. Use 3.14 for . Round your answer to the nearest tenth.

A = • r • r

9 ft

First, find the area of the whole circle with the radius of 9 ft.

A ≈ 3.14 • 9 ft • 9 ft Do not round yet.A ≈ 254.34 ft2



Finding the Area of a Semicircle

Slide 8.6- 112

9 ftNow, divide the area of the whole

circle by 2.

2254.34 ft2

127.17 ft2=

The last step is to round it the nearest tenth.

The area of the semicircle is approximately 127.2 ft2.


ParallelExample 5 Applying the Concept of Circumference

Slide 8.6- 113

A circular rug is 10 feet in diameter. The cost of fringe for the edge is $3.20 per foot. What will it cost to add fringe to the rug? Use 3.14 for .

C = 3.14 • 10 ftC ≈ 31.4 ft

C d

cost = cost per foot • circumference

cost = $3.20 31.4 ft1 ft 1•

cost = $100.48The cost of adding fringe to the rug is $100.48.


ParallelExample 7 Using Prefixes to Understand Math Terms

Slide 8.6- 114

Listed below are some Latin and Greek root words and prefixeswith their meanings in parentheses. List at least one math termand one nonmathematical word that use each prefix or root word.

cent- (100): centigram; centipede

circum- (around):

de- (down):

dec- (10):

circumference; circumvent

decrease; defame

decagon; decathlon


10.6/10.7 VolumeObjectives

Slide 8.7- 115

Find the volume of a 1. rectangular solid;2. sphere;3. cylinder;4. cone and pyramid.


Slide 8.7- 116

A shoe box and a cereal box are examples of three-dimensional (or solid) figures.

The three dimensions are length, width, and height.

If you want to know how much a shoe box will hold, you find its volume.


Slide 8.7- 117

Three sizes of cubic units


Slide 8.7- 118


Find the volume of each box.a.The figure is made up of3 layers of 20 cubes each,so its volume is 60 cubic

centimeters (cm3).

ParallelExample 1 Finding the Volume of Rectangular

Solids

Slide 8.7- 119

5 cm4 cm

3 cm

V wl h

4 c5 m c 3 cmm V 360 cmV


Find the volume of each box.b.

ParallelExample 1 Finding the Volume of Rectangular

Solids

Slide 8.7- 120

V wl h

5 in 6 . in 1 n.. 0 iV 3300 in.V

10 in.

6 in. 5 in.


Slide 8.7- 121

A sphere is shown below.Examples of spheres include baseballs, oranges, and Earth.

The radius of a sphere is the distance from the center to the edge of the sphere.


Find the volume of the sphere with the help of calculator. Use 3.14 as the approximate value of . Round to the nearest tenth.

ParallelExample 2 Finding the Volume of a Sphere

Slide 8.7- 122

343

rV

3.14 843

8 8V

8 in.

32143.573 inV 32143. n6 iV


Slide 8.7- 123

Half a sphere is called a hemisphere. The volume of a hemisphere is half the volume of a sphere.


Find the volume of the hemisphere with the help of calculator. Use 3.14 for . Round to the nearest tenth.

ParallelExample 3 Finding the Volume of a Hemisphere

Slide 8.7- 124

323

rV

3.14 623

6 6V

6 in.

3452.2 inV


Slide 8.7- 125

Several cylinders are shown below. The height must be perpendicular to the circular top and bottom of the cylinder.


Find the volume of each cylinder. Use 3.14 for . Round to the nearest tenth. a.

ParallelExample 4 Finding the Volume of Cylinders

Slide 8.7- 126

2V r h

3.1 784 8V 31406.7 mV

16 m

7 m

The diameter is 16 m so the radius is half: 16 ÷ 2 = 8 m.


Find the volume of each cylinder. Use 3.14 for . Round to the nearest tenth. b.

ParallelExample 4 Finding the Volume of Cylinders

Slide 8.7- 127

2V r h

3.14 3 163V 3452.2 ftV

16 ft 3 ft


Slide 8.7- 128

A cone and pyramid are shown below. Notice that the height line is perpendicular to the base in both solids.


Find the volume of the cone. Use 3.14 for . Round to the nearest tenth.

ParallelExample 5 Finding the Volume of a Cone

Slide 8.7- 129

B r r

278.5 cmB

3V B h

First find the value of B in the formula, which is the area of the circular base.

12 cm

5 cm 3.14 5 5B

Now find the volume.

78.53

12V 3314 cm


Slide 8.7- 130


Find the volume of the pyramid. Round to the nearest tenth.

ParallelExample 6 Finding the Volume of a Pyramid

Slide 8.7- 131

5 6B

3V B h

First find the value of B in the formula, which is the area of a rectangular base.

230 cmB Now find the volume.

13

30 2V 3120 cm

12 cm

6 cm5 cm


10.4 (cont.) Pythagorean TheoremObjectives

Slide 8.8- 132

1. Find square roots using the square root key on a calculator.

2. Find the unknown length in a right triangle.

3. Solve application problems involving right triangles.


A number that has a whole number as its square root is called a perfect square.

The first few perfect squares are listed below.


ParallelExample 1 Find the Square Root of Numbers

Slide 8.8- 134

Use a calculator to find each square root. Round answers to the nearest thousandth.

a. 46

The calculator shows 6.782329983; round to 6.782

b. 136The calculator shows 11.66190379; round to 11.662

c. 260The calculator shows 16.1245155; round to 16.125


Slide 8.8- 135

One place you will use square roots is when working with the Pythagorean Theorem. This theorem applies only to right triangles. Recall that a right triangle is a triangle that has one 90° angle. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called legs.




ParallelExample 2 Find the Unknown Length in Right

Triangles

Slide 8.8- 138


a.The unknown length is the side opposite the right angle. Use the formula for finding the hypotenuse.

8 cm

15 cm



= 64 225

= 289 = 17The length is 17 cm. long



Find the Unknown Length in Right Triangles

Slide 8.8- 139


b. Use the formula for finding the leg.

15 ft 2 2leg = hypotenuse leg

= 1600 225 = 1375 37.1

40 ft

2 2leg = 40 15

The length is approximately 37.1 ft long. Copyright © 2010 Pearson Education, Inc. All rights reserved.

ParallelExample 3 Using the Pythagorean Theorem

Slide 8.8- 140

An electrical pole is shown below. Find the length of the guy wire. Round your answer to the nearest tenth of a foot if necessary.



= 1225 3600

= 4825 69.5The length of the guy wire is approximately 69.5 ft.

35ft

60 ft


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Chapter 10 Geometry 2010 Pearson Education, Inc. All rights reserved