9.9 Volume

Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Московский государственный индустриальный университет

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

basic mathematics for college students.pdf

Скачиваний:

147

Добавлен:

10.06.2015

Размер:

15.55 Mб

Скачать

☆

<<< < Предыдущая 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 6566 / 6766 67 > Следующая >>>

REVIEW

65.Write 109 as a percent.

66.Write 78 as a percent.

67.Write 0.827 as a percent.

68.Write 0.036 as a percent.

69.UNIT COSTS A 24-ounce package of green beans sells for $1.29. Give the unit cost in cents per ounce.

S E C T I O N

9.9 STUDY SET

VOCABULARY

Fill in the blanks.

of a three-dimensional ﬁgure is a

801

70.MILEAGE One car went 1,235 miles on 51.3 gallons of gasoline, and another went 1,456 on 55.78 gallons. Which car got the better gas mileage?

71.How many sides does a pentagon have?

72.What is the sum of the measures of the angles of a triangle?

S E C T I O N 9.9

Volume

We have studied ways to calculate the perimeter and the area of two-dimensional ﬁgures that lie in a plane, such as rectangles, triangles, and circles. Now we will consider three-dimensional ﬁgures that occupy space, such as rectangular solids, cylinders, and spheres. In this section, we will introduce the vocabulary associated with these ﬁgures as well as the formulas that are used to ﬁnd their volume. Volumes are measured in cubic units, such as cubic feet, cubic yards, or cubic centimeters. For example,

• We measure the capacity of a refrigerator in cubic feet.

• We buy gravel or topsoil by the cubic yard.

• We often measure amounts of medicine in cubic centimeters.

1 Find the volume of rectangular solids, prisms, and pyramids.

The volume of a three-dimensional ﬁgure is a measure of its capacity. The following illustration shows two common units of volume: cubic inches, written as in.3, and cubic centimeters, written as cm3.

1 cubic inch: 1 in.3

1 cubic centimeter: 1 cm3

1 in.

1 cm

1 in.

1 cm

1 in.

1 cm

The volume of a ﬁgure can be thought of as the number of cubic units that will ﬁt within its boundaries. If we divide the ﬁgure shown in black below into cubes, each cube represents a volume of 1 cm3. Because there are 2 levels with 12 cubes on each level, the volume of the prism is 24 cm3.

Objectives

1Find the volume of rectangular solids, prisms, and pyramids.

2Find the volume of cylinders, cones, and spheres.

1 cm3

2 cm

3 cm

4 cm

802	Chapter 9 An Introduction to Geometry

Self Check 1

How many cubic centimeters are in 1 cubic meter?

Now Try Problem 25

EXAMPLE 1 How many cubic inches are there in 1 cubic foot?

Strategy A ﬁgure is helpful to solve this problem.We will draw a cube and divide each of its sides into 12 equally long parts.

WHY Since a cubic foot is a cube with each side measuring 1 foot, each side also measures 12 inches.

Solution The ﬁgure on the right helps us understand the situation. Note that each level of the cubic foot contains 12 12 cubic inches

and that the cubic foot has 12 levels. We can 1 ft 12 in. use multiplication to count the number of

cubic inches contained in the ﬁgure.There are

12 12 12 1,728

cubic inches in 1 cubic foot. Thus, 1 ft3 1,728 in.3.

Cube	Rectangular Solid

12 in.

1 ft

Sphere

V s3

where s is the length of a side

Prism

				r
h
h
	w
	l
V lwh		V	4	pr3
V lwh		V	3	pr3
			3
where l is the length, w is		where r is the radius

the width, and h is the height

Pyramid

h	h	h	h

V Bh

where B is the area of the base and h is the height

Cylinder

V 13 Bh

where B is the area of the base and h is the height

Cone

V Bh or V pr2h

Bh or V

pr2h

where B is the area of the base,

h is the height, and r is the radius of the base

EXAMPLE 2

In practice, we do not ﬁnd volumes of three-dimensional ﬁgures by counting cubes. Instead, we use the formulas shown in the table on the preceding page. Note that several of the volume formulas involve the variable B. It represents the area of the base of the ﬁgure.

Caution! The height of a geometric solid is always measured along a line perpendicular to its base.

Storage Tanks An oil storage tank is in the form of a rectangular solid with dimensions 17 feet by 10 feet by 8 feet. (See the ﬁgure below.) Find its volume.

9.9 Volume

803

Self Check 2

Find the volume of a rectangular solid with dimensions 8 meters by 12 meters by 20 meters.

Now Try Problem 29

8 ft

10 ft

17 ft

Strategy We will substitute 17 for l , 10 for w, and 8 for h in the formula V lwh and evaluate the right side.

WHY The variable V represents the volume of a rectangular solid.

Solution

V lwh	This is the formula for the volume of a rectangular solid.		5
			170
V 17(10)(8)	Substitute 17 for l, the length, 10 for w, the width, and 8		8
		1,360
1,360	for h, the height of the tank.
	Do the multiplication.

The volume of the tank is 1,360 ft3.

	EXAMPLE 3	Find the volume of the prism
	EXAMPLE 3		10 cm
			10 cm
shown on the right.
Strategy First, we will ﬁnd the area of the base					50 cm
of the prism.
WHY To use the volume formula V Bh, we
need to know B, the area of the prism’s base.
Solution The area of the triangular base of the			6 cm	8 cm
Solution The area of the triangular base of the				8 cm
prism is 21 (6)(8) 24 square centimeters. To ﬁnd
prism is 21 (6)(8) 24 square centimeters. To ﬁnd
its volume, we proceed as follows:
	V Bh	This is the formula for the volume of a triangular prism.			2
					2
					24
	V 24(50)	Substitute 24 for B, the area of the base, and 50 for h,			50
	V 24(50)	Substitute 24 for B, the area of the base, and 50 for h,			1,200
		the height.			1,200
		the height.
1,200		Do the multiplication.

The volume of the triangular prism is 1,200 cm3.

Caution! Note that the 10 cm measurement was not used in the calculation of the volume.

Self Check 3

Find the volume of the prism shown below.

10 in.

12 in.

5 in.

Now Try Problem 33

804	Chapter 9 An Introduction to Geometry

Self Check 4

Find the volume of the pyramid shown below.

20 cm

	cm	16	cm
			cm
12
12

Now Try Problem 37

	EXAMPLE 4	Find the volume of the pyramid
	EXAMPLE 4	Find the volume of the pyramid	shown
on the right.			9 m
Strategy First, we will ﬁnd the area of the square			9 m
Strategy First, we will ﬁnd the area of the square
base of the pyramid.
WHY The volume of a pyramid is 31 of the product of				6 m


the area of its base and its height.			6 m
Solution Since the base is a square with each side			6 m
Solution Since the base is a square with each side

6 meters long, the area of the base is (6 m)2, or 36 m2. To ﬁnd the volume of the pyramid, we proceed as follows:

V	1	Bh	This is the formula for the volume of a pyramid.
	3
V	1	(36)(9)	Substitute 36 for B, the area of the base, and 9 for h, the height.

	3
12(9)			Multiply:	1	(36) 36 12.	1

						12

			3		3	9
108			Complete the multiplication.
						108

The volume of the pyramid is 108 m3.

Self Check 5

Find the volume of the cylinder shown below. Give the exact answer and an approximation to the nearest hundredth.

10 yd

4 yd

Now Try Problem 45

2 Find the volume of cylinders, cones, and spheres.

	EXAMPLE 5	Find the volume of the cylinder shown on the
	EXAMPLE 5	Find the volume of the cylinder shown on the		6 cm
right. Give the exact answer and an approximation to the nearest

hundredth.
hundredth.
Strategy First, we will ﬁnd the radius of the circular base of the
cylinder.					10 cm
cylinder.
WHY To use the formula for the volume of a cylinder, V pr2h,
we need to know r, the radius of the base.
we need to know r, the radius of the base.
Solution Since a
Solution Since a		radius is one-half of the diameter of the	circular base,

r 12 6 cm 3 cm. From the ﬁgure, we see that the height of the cylinder is 10 cm. To ﬁnd the volume of the cylinder, we proceed as follows.

V pr2h	This is the formula for the volume of a cylinder.
V p(3)2(10)	Substitute 3 for r, the radius of the base,
	and 10 for h, the height.
V p(9)(10)	Evaluate the exponential expression: (3)2 9.
90p	Multiply: (9)(10) 90. Write the product so
	that P is the last factor.
282.7433388	Use a calculator to do the multiplication.

The exact volume of the cylinder is 90p cm3. To the nearest hundredth, the volume is 282.74 cm3.

	EXAMPLE 6	Find the volume of the cone shown on
	EXAMPLE 6	Find the volume of the cone shown on
the right. Give the exact answer and an approximation to the
nearest hundredth.			6 ft
Strategy We will substitute 4 for r and 6 for h in the formula			6 ft

V 31 pr2h and evaluate the right side.				4 ft
V 31 pr2h and evaluate the right side.				4 ft

WHY The variable V represents the volume of a cone.

Solution

V	1	pr2h	This is the formula for the volume of a cone.
	3

V	1	p(4)2(6)	Substitute 4 for r, the radius of the base, and 6 for h, the height.
	3

	1	p(16)(6)	Evaluate the exponential expression: (4)2 16.
	3

	2p(16)		Multiply:	1	(6) 2.

			3
	32p		Multiply: 2(16) 32. Write the product so that P is the last factor.
	100.5309649		Use a calculator to do the multiplication.

The exact volume of the cone is 32p ft3. To the nearest hundredth, the volume is 100.53 ft3.

	EXAMPLE 7	Water Towers How many cubic	15 ft
feet of water are needed to ﬁll the spherical water tank			15 ft
feet of water are needed to ﬁll the spherical water tank
shown on the right? Give the exact answer and an
approximation to the nearest tenth.

Strategy We will substitute 15 for r in the formula

V 43 pr3 and evaluate the right side.

WHY The variable V represents the volume of a sphere.

Solution

V	4	pr3		This is the formula for the volume of a sphere.
V	3	pr3		This is the formula for the volume of a sphere.
	3
V	4	p(15)3		Substitute 15 for r, the radius of the sphere.
V	3	p(15)3		Substitute 15 for r, the radius of the sphere.
	3
	4	p (3,375)		Evaluate the exponential expression: (15)3 3,375.

	3
	13,500			Multiply: 4(3,375) 13,500.	1 3 2
	13,500				1 3 2
			p		3375
	3		p		3375
				Divide: 13,500 4,500. Write the product	4
	4,500p					13,500
	4,500p					13,500
				3
				so that P is the last factor.
	14,137.16694			Use a calculator to do the multiplication.

The tank holds exactly 4,500p ft3 of water. To the nearest tenth, this is 14,137.2 ft3.

9.9 Volume

805

Self Check 6

Find the volume of the cone shown below. Give the exact answer and an approximation to the nearest hundredth.

5 mi

2 mi

Now Try Problem 49

Self Check 7

Find the volume of a spherical water tank with radius 7 meters. Give the exact answer and an approximation to the nearest tenth.

Now Try Problem 53

Using Your CALCULATOR Volume of a Silo

A silo is a structure used for storing grain. The silo shown on the right is a cylinder 50 feet tall topped with a dome in the shape of a hemisphere. To ﬁnd the volume of the silo, we add the volume of the cylinder to the volume of the dome.

								1				50 ft
Volumecylinder Volumedome (Areacylinder’s base)(Heightcylinder)									(Volumesphere)
								2	(Volumesphere)
pr2h	1	a	4	pr3b
pr2h		a		pr3b
	2	3		3
	2	3
	2pr											10 ft
pr2h	2pr				Multiply and simplify:		1	14 pr3 2		4 pr3	2pr3 .
pr2h		3			Multiply and simplify:		1	14 pr3 2		4 pr3	2pr3 .
		3					2 3			6	3
p(10)2 (50)					2p(10)3
						Substitute 10 for r and 50 for h.
					3	Substitute 10 for r and 50 for h.
					3

806		Chapter 9 An Introduction to Geometry

We can use a scientiﬁc calculator to make this calculation.

p 10 x2 50 ( 2 p 10 yx 3 ) 3

iStockphoto.com/ VeithSherwood				17802.35837
				17802.35837
	The volume of the silo is approximately 17,802 ft3.
© R.

	ANSWERS TO SELF CHECKS
	ANSWERS TO SELF CHECKS
	1.	1,000,000 cm3 2. 1,920 m3	3. 300 in.3 4. 640 cm3 5. 100p yd3 314.16 yd3
	6.	203 p mi3 20.94 mi3 7. 1,3723	p m3 1,436.8 m3

measure of its capacity.

2. The volume of a ﬁgure can be thought of as the number of units that will ﬁt within its boundaries.

Give the name of each ﬁgure.

3. 4.

5. 6.

7. 8.

CONCEPTS

9.Draw a cube. Label a side s.

10.Draw a cylinder. Label the height h and radius r.

11.Draw a pyramid. Label the height h and the base.

12.Draw a cone. Label the height h and radius r.

13.Draw a sphere. Label the radius r.

14.Draw a rectangular solid. Label the length l , the width w, and the height h.

15.Which of the following are acceptable units with which to measure volume?

ft2	mi3	seconds	days
cubic inches	mm	square yards	in.
pounds	cm2	meters	m3

16.In the ﬁgure on the right, the unit of measurement of length used to draw the ﬁgure is the inch.

a.What is the area of the base of the ﬁgure?

b.What is the volume of the ﬁgure?

17.Which geometric concept (perimeter, circumference, area, or volume) should be applied when measuring each of the following?

a.The distance around a checkerboard

b.The size of a trunk of a car

c.The amount of paper used for a postage stamp

d.The amount of storage in a cedar chest

e.The amount of beach available for sunbathing

f.The distance the tip of a propeller travels

18.Complete the table.

Figure	Volume formula

Cube
Rectangular solid
Prism
Cylinder
Pyramid
Cone
Sphere

19. Evaluate each expression. Leave p in the answer.

a.	1	p(25)6	b.	4	p(125)
	3			3

20.a. Evaluate 13 pr2h for r 2 and h 27. Leave p in the answer.

b.Approximate your answer to part a to the nearest tenth.

NOTATION

21.a. What does “in.3” mean?

b. Write “one cubic centimeter” using symbols.

22.In the formula V 13 Bh, what does B represent?

23. In a drawing, what does the symbol indicate?

24.Redraw the ﬁgure below using dashed lines to show the hidden edges.

9.9 Volume

807

GUIDED PRACTICE

Convert from one unit of measurement to another.

See Example 1.

25.How many cubic feet are in 1 cubic yard?

26.How many cubic decimeters are in 1 cubic meter?

27.How many cubic meters are in 1 cubic kilometer?

28.How many cubic inches are in 1 cubic yard?

Find the volume of each ﬁgure. See Example 2.

29. 30.

7 ft

8 mm

2 ft

4 ft

10 mm

4 mm

31. 32.

		5 in.
		40 ft
		5 in.
	5 in.	40 ft
	5 in.	40 ft
		40 ft

Find the volume of each ﬁgure. See Example 3.

33.	5 cm	34.		13 cm
		0.2 m				0.8 m
						0.8 m
	3 cm	4 cm
			12 cm		5 cm
			12 cm
35.	12 in.	36.
	12 in.	10 in.			24 in.
		10 in.			24 in.
						0.5 ft
						0.5 ft
	2 ft
	2 ft			26 in.
				26 in.
		15 in.
	9 in.

808	Chapter 9 An Introduction to Geometry

Find the volume of each ﬁgure. See Example 4.

37. 38.

21 yd

15 m

7 m

10 yd

7 m

39.

6 ft

2 ft

8 ft

40. 41.

7.0 ft

18 in.

7.2 ft

8.3 ft

13 in.

11 in.

42.

8.0 mm

4.8 mm

9.1 mm

43. 44.

2 yd

11 ft

Area of base

9 yd2

Area of base 33 ft2

Find the volume of each cylinder. Give the exact answer and an approximation to the nearest hundredth. See Example 5.

45.		46.
	4 ft	2 mi
	4 ft
12 ft		6 mi

47. 30 cm

14 cm

48.116 in.

60 in.

Find the volume of each cone. Give the exact answer and an approximation to the nearest hundredth. See Example 6.

49.

13 m

6 m

50.

21 mm

4mm

51.

7 yd

9 yd

52.5 ft

30 ft

Find the volume of each sphere. Give the exact answer and an approximation to the nearest tenth. See Example 7.

53. 54.

6 in.	9 ft

55.	4 cm	56.	10 in.
55.	4 cm	56.	10 in.

TRY IT YOURSELF

Find the volume of each ﬁgure. If an exact answer contains p, approximate to the nearest hundredth.

57.A hemisphere with a radius of 9 inches

(Hint: a hemisphere is an exact half of a sphere.)

58.A hemisphere with a diameter of 22 feet

(Hint: a hemisphere is an exact half of a sphere.)

59.A cylinder with a height of 12 meters and a circular base with a radius of 6 meters

60.A cylinder with a height of 4 meters and a circular base with a diameter of 18 meters

61.A rectangular solid with dimensions of 3 cm by 4 cm by 5 cm

62.A rectangular solid with dimensions of 5 m by 8 m by 10 m

63.A cone with a height of 12 centimeters and a circular base with a diameter of 10 centimeters

64.A cone with a height of 3 inches and a circular base with a radius of 4 inches

65.A pyramid with a square base 10 meters on each side and a height of 12 meters

66.A pyramid with a square base 6 inches on each side and a height of 4 inches

67.A prism whose base is a right triangle with legs 3 meters and 4 meters long and whose height is 8 meters

68.A prism whose base is a right triangle with legs 5 feet and 12 feet long and whose height is 25 feet

9.9 Volume

809

Find the volume of each ﬁgure. Give the exact answer and, when needed, an approximation to the nearest hundredth.

69.

3 cm

70.

10 in.

8 cm

20 in.

8 cm

8 in.

71.16 cm

6 cm

72.

		8 in.
6 in.	4
.	4	in.
3		in.
in
	5 in.

APPLICATIONS

Solve each problem. If an exact answer contains p, approximate the answer to the nearest hundredth.

73.SWEETENERS A sugar cube is 12 inch on each edge. How much volume does it occupy?

74.VENTILATION A classroom is 40 feet long, 30 feet wide, and 9 feet high. Find the number of cubic feet of air in the room.

75.WATER HEATERS Complete the advertisement for the high-efﬁciency water heater shown below.

Over 200 gallons of hot water

from ? cubic feet of space...

27"

17"

810Chapter 9 An Introduction to Geometry

76.REFRIGERATORS The largest refrigerator advertised in a JC Penny catalog has a capacity of 25.2 cubic feet. How many cubic inches is this?

77.TANKS A cylindrical oil tank has a diameter of

6 feet and a length of 7 feet. Find the volume of the tank.

78.DESSERTS A restaurant serves pudding in a conical dish that has a diameter of 3 inches. If the dish is 4 inches deep, how many cubic inches of pudding are in each dish?

79.HOT-AIR BALLOONS The lifting power of a spherical balloon depends on its volume. How many cubic feet of gas will a balloon hold if it is 40 feet in diameter?

80.CEREAL BOXES A box of cereal measures

3 inches by 8 inches by 10 inches. The manufacturer plans to market a smaller box that measures 212 by 7 by 8 inches. By how much will the volume be

reduced?

81.ENGINES The compression ratio of an engine is the volume in one cylinder with the piston at bottom- dead-center (B.D.C.), divided by the volume with the piston at top-dead-center (T.D.C.). From the data given in the following ﬁgure, what is the compression ratio of the engine? Use a colon to express your answer.

Volume before

compression: 30.4 in.3

B.D.C.

Volume after

compression: 3.8 in.3

T.D.C.

82. GEOGRAPHY Earth is not a perfect sphere but is slightly pear-shaped. To estimate its volume, we will assume that it is spherical, with a diameter of about 7,926 miles. What is its volume, to the nearest one billion cubic miles?

83. BIRDBATHS

30 in.

a.The bowl of the birdbath shown on the right is in the shape of a hemisphere (half of a sphere). Find its volume.

b.If 1 gallon of water occupies 231 cubic inches of space, how many gallons of water does the birdbath hold? Round to the nearest tenth.

84.CONCRETE BLOCKS Find the number of cubic inches of concrete used to make the hollow, cubeshaped, block shown below.

5 in.

8 in.

WRITING

85.What is meant by the volume of a cube?

86.The stack of 3 5 index cards shown in ﬁgure (a) forms a right rectangular prism, with a certain volume. If the stack is pushed to lean to the right, as in ﬁgure (b), a new prism is formed. How will its volume compare to the volume of the right rectangular prism? Explain your answer.

(a)

(b)

87.Are the units used to measure area different from the units used to measure volume? Explain.

88.The dimensions (length, width, and height) of one rectangular solid are entirely different numbers from the dimensions of another rectangular solid. Would it be possible for the rectangular solids to have the same volume? Explain.

REVIEW

89.Evaluate: 5(5 2)2 3

90.BUYING PENCILS Carlos bought 6 pencils at $0.60 each and a notebook for $1.25. He gave the clerk a $5 bill. How much change did he receive?

91.Solve: x 4

92.38 is what percent of 40?

93.Express the phrase “3 inches to 15 inches” as a ratio in simplest form.

94.Convert 40 ounces to pounds.

95.Convert 2.4 meters to millimeters.

96.State the Pythagorean equation.

Chapter 9 Summary and Review

811

STUDY SKILLS CHECKLIST

Know the Vocabulary

A large amount of vocabulary has been introduced in Chapter 9. Before taking the test, put a checkmark in the box if you can deﬁne and draw an example of each of the given terms.

Point, line, plane

Line segment, midpoint

Ray, angle, vertex

Acute angle, obtuse angle, right angle, straight angle

Adjacent angles, vertical angles

Complementary angles, supplementary angles

Congruent segments, congruent angles

Parallel lines, perpendicular lines, a transversal

Alternate interior angles, interior angles, corresponding angles

Polygon, triangle, quadrilateral, pentagon, hexagon, octagon

Equilateral triangle, isosceles triangle, scalene triangle

Acute triangle, obtuse triangle

Right triangle, hypotenuse, legs

Congruent triangles, similar triangles

Parallelogram, rectangle, square, rhombus, trapezoid, isosceles trapezoid

Circle, arc, semicircle, radius, diameter

Rectangular solid, cube, sphere, prism, pyramid, cylinder, cone

C H A P T E R 9 SUMMARY AND REVIEW

S E C T I O N 9.1 Basic Geometric Figures; Angles

DEFINITIONS AND CONCEPTS

The word geometry comes from the Greek words geo (meaning Earth) and metron (meaning measure).

Geometry is based on three undeﬁned words: point, line, and plane.

EXAMPLES

Point

Line BC

Points are	We can name a
labeled with	line using any
capital letters.	two points on it.

Plane EFG

E G

Floors, walls, and table tops are all parts of planes.

812

Chapter 9 An Introduction to Geometry

A line segment is a part of a line with two endpoints.

Line segment

Ray CD

Every line segment has a midpoint, which divides the

B endpoint

segment into two parts of equal length.

) m(

)

AM MB

midpoint

The notation m(AM) is read as “the measure of line

segment AM.”

endpoint

When two line segments have the same measure, we say that they are congruent. Read the symbol as “is congruent to.”

A ray is a part of a line with one endpoint.

An angle is a ﬁgure formed by two rays (called sides)	The angle below can be written as BAC, CAB, A, or 1.
with a common endpoint. The common endpoint is					B
called the vertex of the angle.	Angle				B
called the vertex of the angle.	Angle						Sides of
We read the symbol as “angle.”	A		1				Sides of
We read the symbol as “angle.”	A		1				the angle
	Vertex of						the angle
	Vertex of				C
	the angle				C

When two angles have the same measure, we say that
they are congruent.		D					S
A protractor is used to ﬁnd the measure of an angle.	Congruent
A protractor is used to ﬁnd the measure of an angle.	angles
One unit of measurement of an angle is the degree.			60°			60°
The notation m( DEF) is read as “the measure of	E			F			T
DEF.”				F			V
DEF.”	Since m( DEF) m( STV), we say that DEF STV.
	Since m( DEF) m( STV), we say that DEF STV.

An acute angle has a measure that is greater than 0°
but less than 90°. An obtuse angle has a measure that						180°
is greater than 90° but less than 180°.A straight angle			130°			180°
is greater than 90° but less than 180°.A straight angle	40°		130°
measures 180°.	40°
	Acute angle		Obtuse angle				Straight angle

A right angle measures 90°.
	Right angle				A		symbol is
					A		symbol is
					often used to
			90°		label a right angle.
			90°


Two angles that have the same vertex and are side-	Two angles with degree measures of x
Two angles that have the same vertex and are side-	Two angles with degree measures of x						Adjacent angles
by-side are called adjacent angles.	and 21° are adjacent angles, as shown
	here. Use the information in the ﬁgure to
	ﬁnd x.						21°
							21°
							32°
							x

We can use the algebra concepts of variable and	The sum of the measures of the two adjacent angles is 32°:
equation to solve many types of geometry problems.	x 21° 32°				The word sum indicates addition.
	x 21° 32°				The word sum indicates addition.
	x 21° 21° 32° 21°				Subtract 21° from both sides.
	x 11°				Do the subtraction.
	Thus, x is 11°.

Chapter 9 Summary and Review	813

When two lines intersect, pairs of nonadjacent angles	Vertical angles
are called vertical angles.

Vertical angles are congruent (have the same Refer to the ﬁgure below. Find x and m( XYZ).
measure).			Z
X	3x +	20°

	Y
R	2x + 70°		T

Since the angles are vertical angles, they have equal measures.
3x 20° 2x 70°		Set the expressions equal.
3x 20° 2x 2x 70° 2x Eliminate 2x from the right
			side.
x 20° 70°		Combine like terms.
x 50°		Subtract 20° from both sides.

Thus, x is 50°. To ﬁnd m( XYZ), evaluate the expression 3x 20° for x 50°.

	3x 20° 3(50°) 20°	Substitute 50° for x.
	150° 20°	Do the multiplication.
	170°	Do the addition.
	Thus, m( XYZ) 170°.
If the sum of two angles is 90°, the angles are	Complementary angles	Supplementary angles
complementary.	63° 27° 90°	146° 34° 180°
	63° 27° 90°	146° 34° 180°
If the sum of two angles is 180°, the angles are
supplementary.
	63°	146°
		146°
	27°	34°

We can use algebra to ﬁnd the complement of an	Find the complement of an 11° angle.
angle.	Let x the measure of the complement (in degrees).
	Let x the measure of the complement (in degrees).
	x 11° 90° The sum of the angles’ measures must be 90°.
	x 79° To isolate x, subtract 11° from both sides.
	The complement of an 11° angle has measure 79°.

We can use algebra to ﬁnd the supplement of an	Find the supplement of a 68° angle.
angle.	Let x the measure of the supplement (in degrees).
	Let x the measure of the supplement (in degrees).
	x 68° 180° The sum of the angles’ measures must be 180°.
	x 112° To isolate x, subtract 68° from both sides.
	The supplement of a 68° angle has measure 112°.

814	Chapter 9 An Introduction to Geometry

REVIEW EXERCISES

1.In the illustration, give the name of a point, a line, and a plane.

G H

2.a. In the ﬁgure below, ﬁnd m(AG).

b.Find the midpoint of BH.

c.Is AC GE?

A B C D E F G H

3. Give four ways to name the angle shown below.

4.a. Is the angle shown above acute or obtuse?

b.What is the vertex of the angle?

c.What rays form the sides of the angle?

d.Use a protractor to ﬁnd the measure of the angle.

5.Identify each acute angle, right angle, obtuse angle, and straight angle in the ﬁgure below.

E		D
E
	2
1		90°
A	B	C

6.In the ﬁgure above, is ABD CBD?

¡¡

7.In the ﬁgure above, are AC and AB the same ray?

8.The measures of several angles are given below. Identify each angle as an acute angle, a right angle, an obtuse angle, or a straight angle.

a.m( A) 150°

b.m( B) 90°

c.m( C) 180°

d.m( D) 25°

9. The two angles shown here are

adjacent angles. Find x.

50°

35°

10. Line AB is shown in the ﬁgure below. Find y.

y	30°

A	B

11. Refer to the ﬁgure on the right.

	a. Find m( 1).	2
	a. Find m( 1).	1
	65°	1

b.Find m( 2).

12.Refer to the ﬁgure below.

a.What is m( ABG)?

b.What is m( FBE)?

c.What is m( CBD)?

d.What is m( FBG)?

e.Are CBD and DBE complementary angles?

	C
		D
	39°
A	B	E
G
	F
13. Refer to the ﬁgure.	E	5x + 25°	G
a. Find x.		F
		F
b. What is m( HFI)?	I	6x + 5°	H
c. What is m( GFH)?	I	6x + 5°	H
c. What is m( GFH)?

14.Find the complement of a 71° angle.

15.Find the supplement of a 143° angle.

16.Are angles measuring 30°, 60°, and 90° supplementary?

Chapter 9 Summary and Review

815

S E C T I O N 9.2 Parallel and Perpendicular Lines

DEFINITIONS AND CONCEPTS	EXAMPLES
If two lines lie in the same plane, they are called coplanar.	Parallel lines	Perpendicular lines
Parallel lines are coplanar lines that do not intersect.
Parallel lines are coplanar lines that do not intersect.
We read the symbol as “is parallel to.”
Perpendicular lines are lines that intersect and form right
angles.
angles.
We read the symbol as “is perpendicular to.”

A line that intersects two coplanar lines in two distinct

Transversal

Corresponding angles

(different) points is called a transversal.

• 1 5

When a transversal intersects two coplanar lines, four

• 2 6

• 3 7

pairs of corresponding angles are formed.

• 4 8

If two parallel lines are cut by a transversal, corresponding

angles are congruent (have equal measures).

When a transversal intersects two coplanar lines, two

Transversal

Alternate interior angles

pairs of interior angles and two pairs of alternate interior

• 1 4

angles are formed.

• 2 3

If two parallel lines are cut by a transversal, alternate

Interior angles

interior angles are congruent (have equal measures).

m( 1)

m( 3) = 180°

If two parallel lines are cut by a transversal, interior angles

m( 2) m( 4) = 180°

on the same side of the transversal are supplementary.

We can use algebra to ﬁnd the unknown measures of

In the ﬁgure, l1 l2. Find x

corresponding angles.

and the measure of each

5x + 15°

angle that is labeled.

Since the lines are parallel,

4x + 35°

and the angles are

corresponding angles, the

angles are congruent.

5x 15° 4x 35° The angle measures are equal.

x 15° 35° Subtract 4x from both sides.

x 20° To isolate x, subtract 15° from both sides.

Thus, x is 20°. To ﬁnd the measures of the angles labeled in

the ﬁgure, we evaluate each expression for x 20°.

5x 15° 5(20°) 15°

4x 35° 4(20°) 35°

100° 15°

80° 35°

115°

The measure of each angle is 115°.

816	Chapter 9 An Introduction to Geometry


	We can use algebra to ﬁnd the unknown	In the ﬁgure, l1 l2. Find x and the
	measures of interior angles.	measure of each angle that is labeled.			l1
					l1
		Since the angles are interior angles		4x + 17°
		on the same side of the transversal,		x – 12°
		they are supplementary.			l2
					l2

		4x 17° x 12° 180°
		4x 17° x 12° 180°	The sum of the measures of two
			supplementary angles is 180°.
		5x 5° 180°	Combine like terms.
		5x 175°	Subtract 5° from both sides.
		x 35°	Divide both sides by 5.
		Thus, x is 35°. To ﬁnd the measures of the angles in the ﬁgure, we
		evaluate the expressions for x 35°.
		4x 17° 4(35°) 17°	x 12° 35° 12°
		4x 17° 4(35°) 17°	x 12° 35° 12°
		140° 17°	23°
		157°
		The measures of the angles labeled in the ﬁgure are 157° and 23°.

REVIEW EXERCISES

17.a. Lines l1 and l2 shown in ﬁgure (a) below do not intersect and are coplanar. What word describes the lines?

b.In ﬁgure (a), line l3 intersects lines l1 and l2 in two distinct (different) points. What is the name given to line l3?

c.What word describes the two lines shown in ﬁgure (b) below?

l1 l2

(a)

(b)

18.Identify all pairs of alternate interior angles shown in the ﬁgure below.

8 7

5 6

4 3

1 2

19.Refer to the ﬁgure in Problem 18. Identify all pairs of corresponding angles.

20.Refer to the ﬁgure in Problem 18. Identify all pairs of vertical angles.

21.In the ﬁgure below, l1 l2. Find the measure of each angle.

110°

22. In the ﬁgure on the right,

. Find the measure

70°

each angle that is labeled.

60°

50°

23. In the ﬁgure below, l1 l2.

a.Find x.

b.Find the measure of each angle that is labeled.

2x − 30°

x + 10°

24.In the ﬁgure below, l1 l2.

a.Find x.

b.Find the measure of each angle that is labeled.

3x + 50°

4x − 10°

25.In the ﬁgure below, AB DC.

a.Find x.

b.Find the measure of each angle that is labeled.

A		B
	7x − 46°

D	2x + 9°	C

Chapter 9 Summary and Review

817

26.In the ﬁgure below, EF HI.

a.Find x.

b.Find the measure of each angle that is labeled.

E	G	5x − 33°
	3x + 13°	I
	3x + 13°
F

S E C T I O N 9.3 Triangles

DEFINITIONS AND CONCEPTS

A polygon is a closed geometric ﬁgure with at least three line segments for its sides. The points at which the sides intersect are called vertices. A regular polygon has sides that are all the same length and angles that are all the same measure.

The number of vertices of a polygon is equal to the number of sides it has.

Classifying Polygons

Number	Name of	Number	Name of
of sides	polygon	of sides	polygon

3	triangle	8	octagon
4	quadrilateral	9	nonagon
5	pentagon	10	decagon
6	hexagon	12	dodecagon

A triangle is a polygon with three sides (and three vertices).

Triangles can be classiﬁed according to the lengths of their sides.

Tick marks indicate sides that are of equal length.

EXAMPLES

Polygon			Regular polygon
vertex
side	side
vertex		vertex
		vertex
side		side
side
vertex	side	vertex
vertex	side
Quadrilateral		Hexagon	Octagon
(4 sides)		(6 sides)	(8 sides)

Equilateral triangle	Isosceles triangle	Scalene triangle
(all sides of	(at least two sides of	(no sides of
equal length)	equal length)	equal length)

818		Chapter 9 An Introduction to Geometry

Triangles can be classiﬁed by their angles.

Acute triangle	Obtuse triangle	Right triangle
(has three acute angles)	(has an obtuse angle)	(has one right angle)

The longest side of a right triangle is called the		Right triangle
hypotenuse, and the other two sides are called legs.		Hypotenuse
The hypotenuse of a right triangle is always opposite		Hypotenuse
the 90° (right) angle. The legs of a right triangle are		(longest side)
the 90° (right) angle. The legs of a right triangle are	Leg
adjacent to (next to) the right angle.		Leg
adjacent to (next to) the right angle.

In an isosceles triangle, the angles opposite the sides		Isosceles triangles
of equal length are called base angles. The third	Vertex angle
angle is called the vertex angle. The third side is	Vertex angle
angle is called the vertex angle. The third side is
called the base.
Isosceles Triangle Theorem: If two sides of a
triangle are congruent, then the angles opposite	Base angle	Base angle
those sides are congruent.	Base angle	Base angle
those sides are congruent.
Converse of the Isosceles Triangle Theorem: If two	Base
angles of a triangle are congruent, then the sides
opposite the angles have the same length, and the
triangle is isosceles.

The sum of the measures of the angles of any	Find the measure of each angle of ABC.					B
triangle is 180°.
We can use algebra to ﬁnd unknown angle	The sum of the angle measures of any				3x – 25°
measures of a triangle.	triangle is 180°:
	x 3x 25° x 5° 180°				x – 5°	x
	x 3x 25° x 5° 180°				C	A
	5x 30° 180°			Combine like terms.
		5x 210°		Add 30° to both sides.
		x 42°		Divide both sides by 5.
	To ﬁnd the measures of B and C, we evaluate the expressions
	3x 25° and x 5° for x 42°.
	3x 25° 3(42°) 25°		x 5° 42° 5°
	3x 25° 3(42°) 25°		x 5° 42° 5°
	126° 25°			37°
	101°
	Thus, m( A) 42°, m( B) 101°, and m( C) 37°.

We can use algebra to ﬁnd unknown angle measures	If the vertex angle of an isosceles triangle measures 26°, what is the
of an isosceles triangle.	measure of each base angle?
of an isosceles triangle.	measure of each base angle?				26°
	If we let x represent the measure of				26°
	If we let x represent the measure of				x	x
	one base angle, the measure of the				x	x


	other base angle is also x. (See the ﬁgure.) Since the sum of the
	measures of the angles of any triangle is 180°, we have
	x x 26° 180°
	2x 26° 180° On the left side, combine like terms.
	2x 154°	To isolate 2x, subtract 26° from both sides.
	x 77°	To isolate x, divide both sides by 2.
	The measure of each base angle is 77°.

REVIEW EXERCISES

27.For each of the following polygons, give the number of sides it has, tell its name, and then give the number of vertices that it has.

a. b.

c. d.

e. f.

28.Classify each of the following triangles as an equilateral triangle, an isosceles triangle, a scalene triangle, or a right triangle. Some ﬁgures may be correctly classiﬁed in more than one way.

a.	b.
	6 cm	7 cm
8 in.	8 in.
		9 cm
c.	d.
5 m	5 m	44°
5 m		44°

29.Classify each of the following triangles as an acute, an obtuse, or a right triangle.

a. b.

90°

70° 20°

50° 50°

c. 160°

15° 5°

60°

50°

Chapter 9 Summary and Review

819

30. Refer to the triangle shown here.			Y
a.	What is the measure of X?
b.	What type of triangle is it?
c.	What two line segments	Z	X
c.	What two line segments

are the legs?

d.What line segment is the hypotenuse?

e.Which side of the triangle is the longest?

f.Which side is opposite X?

In each triangle shown below, ﬁnd x.

31. 32. x

70°

70° 20°

60° x

33.In ABC, m( B) 32° and m( C) 77°. Find m( A).

34.For the triangle shown below, ﬁnd x. Then determine the measure of each angle of the triangle.

x + 10°

5x + 26°

35.One base angle of an isosceles triangle measures 65°. Find the measure of the vertex angle.

36.The measure of the vertex angle of an isosceles triangle is 68°. Find the measure of each base angle.

37. Find the measure of C	A
of the triangle shown	56.5°
here.		C
	B
38. Refer to the ﬁgure shown	D	E
here. Find m( C).	D
here. Find m( C).
	81°	19°
		19°
	50°
	47°
A	B	C

70°

820 Chapter 9 An Introduction to Geometry

S E C T I O N 9.4

The Pythagorean Theorem

DEFINITIONS AND CONCEPTS

EXAMPLES

Pythagorean theorem

Find the length of the hypotenuse

If a and b are the lengths of the legs of a right triangle

of the right triangle shown here.

and c is the length of the hypotenuse, then

We will let a 6 and b 8, and

6 in.

a2 b2 c2

substitute into the Pythagorean

Hypotenuse

equation to ﬁnd c.

8 in.

Leg

a2 b2 c2

This is the Pythagorean equation.

62 82 c2

Substitute 6 for a and 8 for b.

36 64 c2

Evaluate the exponential expressions.

100 c2

Do the addition.

Leg

a2 b2 c2 is called the Pythagorean equation.

c2 100

Reverse the sides of the equation so that c2 is

on the left.

To ﬁnd c, we must ﬁnd a number that, when squared, is 100.There

are two such numbers, one positive and one negative; they are the

square roots of 100. Since c represents the length of a side of a

triangle, c cannot be negative. For this reason, we need only ﬁnd

the positive square root of 100 to get c.

c 1

The symbol 1

is used to indicate the postive

100

square root of a number.

c 10

Because 102 100.

The length of the hypotenuse of the triangle is 10 in.

When we use the Pythagorean theorem to ﬁnd the

The lengths of two sides of a right triangle are

length of a side of a right triangle, the solution is

shown here. Find the missing side length.

9 ft

11 ft

sometimes the square root of a number that is not a

We may substitute 9 for either a or b, but 11

perfect square. In that case, we can use a calculator to

must be substituted for the length c of the

approximate the square root.

hypotenuse. If we substitute 9 for a, we can ﬁnd the unknown side

length b as follows.

a2 b2 c2

This is the Pythagorean equation.

92 b2 112

Substitute 9 for a and 11 for c.

81 b2 121

Evaluate each exponential expression.

81 b2 81 121 81

To isolate b2 on the left side,

subtract 81 from both sides.

b2 40

We must ﬁnd a number that, when squared, is 40. Since b

represents the length of a side of a triangle, we consider only the

positive square root.

This is the exact length.

The missing side length is exactly

feet long. Since 40 is not a

perfect square, we use a calculator to approximate

To the

40.

nearest hundredth, the missing side length is 6.32 ft.

Chapter 9 Summary and Review	821

The converse of the Pythagorean theorem:

If a triangle has sides of lengths a, b, and c, such that a2 b2 c2, then the triangle is a right triangle.


Is the triangle shown here a right triangle?
We must substitute the longest side length,			12 cm	8 cm
12, for c, because it is the possible				8 cm
12, for c, because it is the possible
hypotenuse.The lengths of 8 and 10 may be
substituted for either a or b.			10 cm
a2 b2	c2	This is the Pythagorean equation.
82 102	122	Substitute 8 for a, 10 for b, and 12 for c.
64 100	144	Evaluate each exponential expression.
164	144	This is a false statement.

Since 164 144, the triangle is not a right triangle.

REVIEW EXERCISES

Refer to the right triangle below.

39.Find c, if a 5 cm and b 12 cm.

40.Find c, if a 8 ft and b 15 ft.

41.Find a, if b 77 in. and c 85 in.

42.Find b, if a 21 ft and c 29 ft.

The lengths of two sides of a right triangle are given. Find the missing side length.Give the exact answer and an approximation to the nearest hundredth.

43.

16 m

5 m

44.

30 in.

20in.

45.HIGH-ROPES ADVENTURE COURSES A builder of a high-ropes adventure course wants to secure a pole by attaching a support cable from the anchor stake 55 inches from the pole’s base to a

point 48 inches up the pole. See the illustration in the next column. How long should the cable be?

Support cable

48 in.

55 in.

46.TV SCREENS Find the height of the television screen shown. Give the exact answer and an approximation to the nearest inch.

41in.

52 in.

Determine whether each triangle shown here is a right triangle.

47. 48. 9

11	2
11
8	7
	7
15

822 Chapter 9 An Introduction to Geometry

S E C T I O N 9.5 Congruent Triangles and Similar Triangles

DEFINITIONS AND CONCEPTS

EXAMPLES

If two triangles have the same size and the same

shape, they are congruent triangles.

ABC DEF

Corresponding parts of congruent triangles are

There are six pairs of congruent parts: three pairs of congruent

congruent (have the same measure).

angles and three pairs of congruent sides.

• m( A) m( D)

• m(

) m(

)

• m( B) m( E)

• m(

) m(

)

• m( C) m( F)

• m(

) m(

)

Three ways to show that two triangles are

MNO RST by the SSS property.

congruent are:

1.The SSS property If three sides of one triangle are congruent to three sides of a second triangle,

6 in.

4 in.

6 in.

the triangles are congruent.

MO RT

NO ST

7 in.

The SAS property If two sides and the angle

DEF XYZ by the SAS property.

between them in one triangle are congruent,

respectively, to two sides and the angle between

them in a second triangle, the triangles are

congruent.

5 ft

2 ft

DF XZ

92°

D X

5 ft

DE XY

2 ft

The ASA property If two angles and the side

ABC TUV by the ASA property.

between them in one triangle are congruent,

respectively, to two angles and the side between

them in a second triangle, the triangles are

10 m

A T

135°

congruent.

135°

20°

AB TU

10 m

B U

Similar triangles have the same shape, but not

EFG WXY by the AAA similarity theorem.

necessarily the same size.

15°

We read the symbol as “is similar to.”

AAA similarity theorem

15°

If the angles of one triangle are congruent to

corresponding angles of another triangle, the

25°

E W

triangles are similar.

140°

25°

F X

G Y

Chapter 9 Summary and Review	823

Property of similar triangles

If two triangles are similar, all pairs of corresponding sides are in proportion.

			Similar triangles are
			determined by the
			tree and its shadow
			and the man and his
		h	shadow. Since the
			triangles are similar,
	5 ft		the lengths of their
			corresponding sides
3 ft	27 ft		are in proportion.
3 ft	27 ft

LANDSCAPING A tree casts a shadow 27 feet long at the same time as a man 5 feet tall casts a shadow 3 feet long. Find the height of the tree.

If we let h the height of the tree, we can ﬁnd h by solving the following proportion.

The height of the tree

The length of the tree’s shadow

The height of the man

The length of the man’s shadow

5(27)

Find each cross product and set them equal.

135

Do the multiplication.

135

To isolate h, divide both sides by 1.3.

Do the division.

The tree is 45 feet tall.

REVIEW EXERCISES

49.Two congruent triangles are shown below. Complete the list of corresponding parts.

a.		A corresponds to		.
b.		B corresponds to		.
c.		C corresponds to		.
			corresponds to	.
d.		AC
			corresponds to	.
e.		AB
			corresponds to	.
f.		BC
			C	F
	A		B E	D

50.Refer to the ﬁgure below, where ABC XYZ. a. Find m( X).

52.

70° 70°

53.

70°			70°
60°		50°	60°	50°
54.
50°		60°	50°	60°
	6 cm		6 cm
Determine whether the triangles are similar.
55.				56.	35°
					35°
	50°	50°
50°		50°
				35°

b.	Find m( C).
	Find m(		).
c.	Find m(	YZ	).		X
d. Find m(				).
d. Find m(		AC		).
				C	9 in.
					9 in.
				6 in.
	32°				61°	Z
						Z
A				B	Y
A				B	Y

Determine whether the triangles in each pair are congruent. If they are, tell why.

51.

3 in.	3 in. 3 in.	3 in.
3 in.		3 in.

57. In the ﬁgure below, RST MNO. Find x and y.

	R	N
		N
	16	x
	16
	32	M
S	32	7
S
	y	8
	y

58.HEIGHT OF A TREE A tree casts a 26-foot shadow at the same time a woman 5 feet tall casts a 2-foot shadow. What is the height of the tree? (Hint: Draw a diagram ﬁrst and label the side lengths of the similar triangles.)

824 Chapter 9 An Introduction to Geometry

S E C T I O N 9.6

Quadrilaterals and Other Polygons

DEFINITIONS AND CONCEPTS

EXAMPLES

A quadrilateral is a polygon with four sides. Use the

Quadrilateral WXYZ

capital letters that label the vertices of a quadrilateral

to name it.

Diagonal XZ

A segment that joins two nonconsecutive vertices of

Diagonal WY

a polygon is called a diagonal of the polygon.

Some special types of quadrilaterals are shown on the

right.

Parallelogram

Rectangle

Square

(Opposite sides

(Parallelogram with

(Rectangle with

parallel)

four right angles)

sides of equal length)

			Rhombus		Trapezoid
			(Parallelogram with		(Exactly two
			sides of equal length)		sides parallel)
A rectangle is a quadrilateral with four right angles.			Rectangle ABCD
			A	30 in.	B
			A		B
					17 in. 16 in.
				E
Properties of rectangles:			D		C
Properties of rectangles:
1.	All four angles are right angles.	1.	m( DAB) m( ABC) m( BCD) m( CDA) 90°
2.	Opposite sides are parallel.	2.	AD BC and AB DC
3.	Opposite sides have equal length.	3.	m(AD) 16 in. and m(DC) 30 in.
4.	Diagonals have equal length.	4.	m(DB) m(AC) 34 in.
5.	The diagonals intersect at their midpoints.	5.	m(DE) m(AE) m(EC) 17 in.
Conditions that a parallelogram must meet to ensure		Read Example 2 on page 769 to see how these two conditions are
that it is a rectangle:		used in construction to “square a foundation.”
1.	If a parallelogram has one right angle, then the		A	12 ft	B
	parallelogram is a rectangle.		A	12 ft	B
	parallelogram is a rectangle.				9 ft
			9 ft		9 ft
2.	If the diagonals of a parallelogram are congruent,		9 ft
	then the parallelogram is a rectangle.		D	12 ft	C
			D	12 ft	C

A trapezoid is a quadrilateral with exactly two sides parallel.

The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs.

If the legs (the nonparallel sides) of a trapezoid are of equal length, it is called an isosceles trapezoid.

In an isosceles trapezoid, both pairs of base angles are congruent.

The sum S, in degrees, of the measures of the angles of a polygon with n sides is given by the formula

S (n 2)180°

We can use the formula S (n 2)180° to ﬁnd the number of sides a polygon has.

			Chapter 9 Summary and Review			825
		Trapezoid ABCD
	A	Upper base		B
		Upper base
			angles
	Leg		angles	Leg	AB \|\| DC
					AB \|\| DC


			Lower base
			angles
	D		Lower base		C
	D

Find the sum of the angle measures of a hexagon.
Since a hexagon has 6 sides,we will substitute 6 for n in the formula.
S (n 2)180°
S (6 2)180° Substitute 6 for n, the number of sides.
(4)180°			Do the subtraction within the parentheses.
720°			Do the multiplication.
The sum of the measures of the angles of a hexagon is 720°.
The sum of the measures of the angles of a polygon is 2,340°. Find
the number of sides the polygon has.
	S (n 2)180°
	2,340° (n 2)180°			Substitute 2,340° for S.
				Now solve for n.
	2,340° 180°n 360°			Distribute the multiplication by 180°.
2,340° 360° 180°n 360° 360° Add 360° to both sides.
	2,700° 180°n			Do the addition.
2,700°		180°n		Divide both sides by 180°.
	180°	180°		Divide both sides by 180°.
	15 n			Do the division.
The polygon has 15 sides.

REVIEW EXERCISES

59.Classify each of the following quadrilaterals as a parallelogram, a rectangle, a square, a rhombus, or a trapezoid. Some ﬁgures may be correctly classiﬁed in more than one way.

2 cm

2 ft

1 ft

60.The length of diagonal AC of rectangle ABCD shown below is 15 centimeters. Find each measure.

a.	m(BD)			D	14 cm	C
b.	m( 1)				40°
c.	m( 2)				E 50°
					2

d.	m(EC)
					1
	m(		)	A		B
e.		AB

61.Refer to rectangle WXYZ below. Tell whether each statement is true or false.

a.m(WX) m(ZY)

b.m(ZE) m(EX)

c.Triangle WEX is isosceles.

d.m(WY) m(WX)

Z		Y
	E
W		X
W		X

826	Chapter 9 An Introduction to Geometry
62. Refer to isosceles trapezoid ABCD below. Find
each measure.					3 yd
a. m( B)				D	3 yd	C
				D		C
					115°
b. m( C)					115°
b. m( C)				4 yd
	m(		)	4 yd
c.	m(	CB	)
				65°		B
				A		B

7 yd

63.Find the sum of the angle measures of an octagon.

64.The sum of the measures of the angles of a polygon is 3,240°. Find the number of sides the polygon has.

S E C T I O N 9.7 Perimeters and Areas of Polygons

DEFINITIONS AND CONCEPTS			EXAMPLES
The perimeter of a polygon is the distance around it.			Find the perimeter of the triangle shown below.

	Figure	Perimeter Formula		23 in.
			11 in.
	Square	P 4s


	Rectangle	P 2l 2w		16 in.
	Triangle	P a b c		16 in.
	Triangle	P a b c
			P a b c	This is the formula for the perimeter of a triangle.
			P a b c	This is the formula for the perimeter of a triangle.
			P 11 16 23 Substitute 11 for a, 16 for b, and 23 for c.
			50	Do the addition.

The perimeter of the triangle is 50 inches.

The area of a polygon is the measure of the amount

Find the area of the triangle shown here.

of surface it encloses.

Figure

Area Formulas

7 m

3 m

Square

A s2

Rectangle

A lw

Parallelogram

A bh

5 m

Triangle

A 21 bh

This is the formula for the area of a triangle.

Trapezoid

A 21 h(b1 b2)

(5)(3)

Substitute 5 for b, the length of the base,

and 3 for h, the height. Note that the side

length 7 m is not used in the calculation.

b a

Write 5 as 5 and 3 as

Multiply the numerators.

Multiply the denominators.

7.5

Do the division.

The area of the triangle is 7.5 m2.

Chapter 9

Summary and Review

827

To ﬁnd the perimeter or area of a polygon, all the

To ﬁnd the perimeter or area

4 ft

measurements must be in the same units. If they are

of the rectangle shown here,

11 in.

not, use unit conversion factors to change them to the

we need to express the length

same unit.

in inches.

4 ft

12 in.

Convert 4 feet to inches using a unit

4 ft

1 ft

conversion factor.

4 12 in.

Remove the common units of feet in the

numerator and denominator. The unit of

inches remain.

48 in.

Do the multiplication.

The length of the rectangle is 48 inches. Now we can ﬁnd the

perimeter (in inches) or area (in in.2) of the rectangle.

If we know the area of a polygon, we can often use

The area of the parallelogram

algebra to ﬁnd an unknown measurement.

shown here is 208 ft2. Find the

height.

26 ft

A bh

This is the formula for the area of a parallelogram.

208 26h

Substitute 208 for A, the area, and 26 for b,

the length of the base.

208

26h

To isolate h, undo the multiplication by 26

by dividing both sides by 26.

8 h

Do the division.

The height of the parallelogram is 8 feet.

To ﬁnd the area of an irregular shape, break up the

Find the area of the shaded ﬁgure

8 cm

shape into familiar polygons. Find the area of each

shown here.

polygon, and then add the results.

We will ﬁnd the area of the lower

8 cm

portion of the ﬁgure (the trapezoid)

and the area of the upper portion

(the square) and then add the results.

18 cm

h(b

b )

This is the formula for the area of a

trapezoid

trapezoid.

Atrapezoid

(10)(8 18)

Substitute 8 for b1, 18 for b2, and

10 for h.

(10)(26)

Do the addition within the parentheses.

130

Do the multiplication.

The area of the trapezoid is 130 cm2.

Asquare s2

This is the formula for the area of a square.

Asquare 82

Substitute 8 for s.

Evaluate the exponential expression.

The area of the square is 64 cm2.

828	Chapter 9 An Introduction to Geometry


		The total area of the shaded ﬁgure is
		Atotal Atrapezoid Asquare
		Atotal 130 cm2 64 cm2
		194 cm2
		The area of the shaded ﬁgure is 194 cm2.

	To ﬁnd the area of an irregular shape, we must	To ﬁnd the area of the shaded ﬁgure below, we subtract the area
	sometimes use subtraction.	of the triangle from the area of the rectangle.

Ashaded Arectangle Atriangle

REVIEW EXERCISES

65.Find the perimeter of a square with sides 18 inches long.

66.Find the perimeter (in inches) of a rectangle that is 7 inches long and 3 feet wide.

Find the perimeter of each polygon.

67.8 m

4 m

6 m

4 m

68.

4 m

8 m

4 m

69.The perimeter of an isosceles triangle is 107 feet. If one of the congruent sides is 24 feet long, how long is the base?

70.a. How many square feet are there in 1 square yard?

b.How many square inches are in 1 square foot?

Find the area of each polygon.

71.3.1 cm

3.1 cm				3.1 cm
3.1 cm				3.1 cm

3.1 cm

72.

50 ft

150 ft

73.

20 ft

15 ft

30 ft

74.

10 in.

40in.

75.12 cm

8 cm

18cm

76.12 ft

14 ft

8 ft

20 ft

77.

4 ft

12 ft

8 ft

20 ft

78.

4 m

10 m

15m

79.The area of a parallelogram is 240 ft2. If the length of the base is 30 feet, what is its height?

80.The perimeter of a rectangle is 48 mm and its width is 6 mm. Find its length.

Chapter 9 Summary and Review

829

81.FENCES A man wants to enclose a rectangular front yard with chain link that costs $8.50 a foot (the price includes installation). Find the cost of enclosing the yard if its dimensions are 115 ft by 78 ft.

82.LAWNS A family is going to have artiﬁcial turf installed in their rectangular backyard that is 36 feet long and 24 feet wide. If the turf costs $48 per square yard, and the installation is free, what will this project cost? (Assume no waste.)

S E C T I O N 9.8 Circles

DEFINITIONS AND CONCEPTS

A circle is the set of all points in a plane that lie a ﬁxed distance from a point called its center. The ﬁxed distance is the circle’s radius.

A chord of a circle is a line segment connecting two points on the circle.

A diameter is a chord that passes through the circle’s center.

Any part of a circle is called an arc.

A semicircle is an arc of a circle whose endpoints are the endpoints of a diameter.

The circumference (perimeter) of a circle is given by the formulas

C pD or C 2pr

where p 3.14159 . . . .

If an exact answer contains p, we can use 3.14 as an approximation, and complete the calculations by hand. Or, we can use a calculator that has a pi key p to ﬁnd an approximation.

EXAMPLES

	A				Arc AB
			Chord		Arc AB
C			Chord		AB
C					AB
	Diameter			CD	B
		O			B
Radius	OE	O
Radius

Semicircle CED

Find the circumference of the circle shown here.

Give the exact answer and an approximation.

8 in.

C 2pr This is the formula for the circumference of a circle.

C 2p(8) Substitute 8 for r, the radius.

C 2(8)p Rewrite the product so that P is the last factor.

C 16p Do the ﬁrst multiplication: 2(8) 16. This is the exact answer.

The circumference of the circle is exactly 16p inches. If we replace p with 3.14, we get an approximation of the circumference.

C 16P

C 16(3.14) Substitute 3.14 for P.

C 50.24 Do the multiplication.

The circumference of the circle is approximately 50.2 inches.

We can also use a calculator to approximate 16p.

C 50.26548246

830

Chapter 9 An Introduction to Geometry

The area of a circle is given by the formula

Find the area of the circle shown here. Give

28 m

A pr2

the exact answer and an approximation to the

nearest tenth.

Since the diameter is 28 meters, the radius is

half of that, or 14 meters.

A pr2

This is the formula for the area of a circle.

A p(14)2

Substitute 14 for r, the radius of the circle.

p(196)

Evaluate the exponential expression.

196p

Write the product so that p is the last factor.

The exact area of the circle is 196p m2. We can use a calculator to

approximate the area.

A 615.7521601 Use a calculator to do the multiplication.

To the nearest tenth, the area is 615.8 m2.

To ﬁnd the area of an irregular shape, break it up into

To ﬁnd the area of the shaded ﬁgure

familiar ﬁgures.

shown here, ﬁnd the area of the triangle

and the area of the semicircle, and then

add the results.

Ashaded figure Atriangle Asemicircle

REVIEW EXERCISES

83. Refer to the ﬁgure.		C	D
a.	Name each chord.

b.	Name each diameter.	A

c.	Name each radius.	O	B

d. Name the center.

84.Find the circumference of a circle with a diameter of 21 feet. Give the exact answer and an approximation to the nearest hundredth.

85.Find the perimeter of the ﬁgure shown below. Round to the nearest tenth.

86.Find the area of a circle with a diameter of 18 inches. Give the exact answer and an approximation to the nearest hundredth.

87.Find the area of the ﬁgure shown in Problem 85. Round to the nearest tenth.

88. Find the area of the shaded		100 in.
region shown on the right.		100 in.


Round to the nearest tenth.

100 in.

10 cm

8 cm

10 cm

					Chapter 9	Summary and Review		831
S E C T I O N 9.9		Volume
S E C T I O N 9.9		Volume


DEFINITIONS AND CONCEPTS				EXAMPLES
The volume of a ﬁgure can be thought of as the				1 cubic inch: 1 in.3 1 cubic centimeter: 1 cm3
number of cubic units that will ﬁt within its boundaries.
Two common units of volume are cubic inches (in.3)					1 in.	1 cm
and cubic centimeters (cm3).						1 cm
					1 in.	1 cm
				1 in.	1 in.
				1 in.

The volume of a solid is a measure of the space it				CARRY-ON LUGGAGE The			Width: 17 in.
occupies.				largest carry-on bag that Alaska
occupies.				largest carry-on bag that Alaska
				Airlines allows on board a ﬂight			Height: 10 in.
	Figure		Volume Formula	is shown on the right. Find the
	Figure		Volume Formula	is shown on the right. Find the
				volume of space that a bag that
	Cube		V s3	volume of space that a bag that		Length: 24 in.
				size occupies.
	Rectangular solid		V lwh	size occupies.
	Rectangular solid		V lwh	V lwh	This is the formula for the volume of a
	Prism		V Bh*	V lwh	This is the formula for the volume of a
	Prism		V Bh*		rectangular solid.
	Pyramid		V 31 Bh*	V 24(17)(10)	Substitute 24 for l, the length, 17 for w, the
	Cylinder		V pr2h		width, and 10 for h, the height of the bag.
	Cone		V 31 pr2h	4,080	Do the multiplication.
	Sphere		V 34 pr3	The volume of the space that the bag occupies is 4,080 in.3.

*B represents the area of the base.

Caution! When ﬁnding the volume of a ﬁgure, only

Find the volume of the prism shown here.

5 ft

use the measurements that

are called for in

the

The area of

the triangular base of the

formula. Sometimes a ﬁgure

may be labeled

with

prism is

9 ft

measurements that are not used.

2 (3)(4) 6 square feet. (The 5-

inch measurement is not used.) To ﬁnd the

volume of the prism, proceed as follows:

3 ft

4 ft

V Bh

This is the formula for the volume of a prism.

V 6(9)

Substitute 6 for B, the area of the base, and 9 for h,

the height.

Do the multiplication.

The volume of the triangular prism is 54 ft3.

The letter B appears in two of the volume formulas.

Find the volume of the pyramid shown here.

It represents the area of the base of the ﬁgure.

Since the base is a square with each side

6 cm

Note that the volume formulas for a pyramid and a

5 centimeters long, the area of the base is

cone contain a factor of 31 .

5 5 25 cm2.

Cone:

V 31 pr2h

5 cm

Pyramid:

5 cm

V 3 Bh

This is the formula for the volume of a pyramid.

Substitute 25 for B, the area of the base, and 6

(25)(6) for h, the height.

25(2)

Multiply the ﬁrst and third factors:

(6) 2.

Complete the multiplication by 25.

The volume of the pyramid is 50 cm3.

832

Chapter 9

An Introduction to Geometry

Note that the volume formulas for a cone, cylinder,

Find the volume of the cylinder shown here.

8 yd

and sphere contain a factor of p.

Give the exact answer and an approximation

Cone

V 1 Pr2h

to the nearest hundredth.

Since a radius is one-half of the diameter of

3 yd

Cylinder

V Pr2h

Sphere

V 34 Pr3

the circular base, r 2 8 yd 4 yd. To ﬁnd

the volume of the cylinder, proceed as follows:

V pr2h

This is the formula for the volume of a cylinder.

V p(4)2(3)

Substitute 4 for r, the radius of the base, and

3 for h, the height.

V p(16)(3)

Evaluate the exponential expression.

48p

Write the product so that P is the last factor.

150.7964474 Use a calculator to do the multiplication.

The exact volume of the cylinder is 48p yd3. To the nearest

hundredth, the volume is 150.80 yd3.

If an exact answer contains p, we can use 3.14 as an

Find the volume of the sphere shown here. Give

approximation, and complete the calculations by hand.

the exact answer and an approximation to the

6 ft

Or, we can use a calculator that has a pi key

to ﬁnd

nearest tenth.

an approximation.

pr3

This is the formula for the

volume of a sphere.

p(6)3

Substitute 6 for r, the radius of the sphere.

p(216)

Evaluate the exponential expression.

864

Multiply: 4(216) 864.

288p

Divide: 864 288.

904.7786842 Use a calculator to do the multiplication.

The volume of the sphere is exactly 288p ft3 .To the nearest tenth,

this is 904.8 ft3.

REVIEW EXERCISES

Find the volume of each ﬁgure. If an exact answer contains P, approximate to the nearest hundredth.

89. 90.

5 cm			8 m
5 cm					6 m
					6 m
		5 cm		10 m
	5 cm

91. 92.

25 mm

5 in.

12 mm

18 mm

16 mm

93. 94.

15 yd

30 in.

20 yd

10 in.

95. 96.

42 m

16 in.

12 m	35 m

Chapter 9 Summary and Review

833

97.FARMING Find the volume of the corn silo shown below. Round to the nearest one cubic foot.

2.5 in.

10 ft

6 in.

16 ft

98.WAFFLE CONES Find the volume of the ice cream cone shown above. Give the exact answer and an approximation to the nearest tenth.

99.How many cubic inches are there in 1 cubic foot?

100.How many cubic feet are there in 2 cubic yards?

834

C H A P T E R 9 TEST

1.Estimate each angle measure. Then tell whether it is an acute, right, obtuse, or straight angle.

2.Fill in the blanks.

a.If ABC DEF, then the angles have the same

		.
b.	Two congruent segments have the same			.
c.	Two different points determine one			.
d.	Two angles are called		if the sum of
	their measures is 90°.

5. Find x. Then ﬁnd m( ABD) and m( CBE).

A	C
3x	2x + 20°
	B
D	E

6.Find the supplement of a 47° angle.

7.Refer to the ﬁgure below. Fill in the blanks.

a. l1 intersects two coplanar lines. It is called a

b.	4 and		are alternate interior angles.
c.	3 and		are corresponding angles.
					l1
					1
				4	2
				4	3
					3
		5		6
		8		7
				7

3.Refer to the ﬁgure below. What is the midpoint of BE?

A		B	C	D		E

2	3	4	5	6	7	8	9

4.Refer to the ﬁgure below and tell whether each statement is true or false.

a.AGF and BGC are vertical angles.

b.EGF and DGE are adjacent angles.

c.m( AGB) m( EGD).

d.CGD and DGF are supplementary angles.

e.EGD and AGB are complementary angles.

8. In the ﬁgure below, l1 l2 and m( 2) 25°. Find the measures of the other numbered angles.

		1	2	l1
	5		6	l1
	5		6
	3	7		l2
4	8			l2
4	8

9. In the ﬁgure below, l1 l2. Find x. Then determine the measure of each angle that is labeled in the ﬁgure.

		l1
	x + 20°	l1
	x + 20°

2x + 10°

10.For each polygon, give the number of sides it has, tell its name, and then give the number of vertices it has.

Chapter 9 Test

835

15. Refer to isosceles trapezoid QRST shown below.

a.	Find m(RS).	b.	Find x.
c.	Find y.	d.	Find z.
	Q	20	R
	z		y
	10
	65°		x
	T	30	S
		30

11.Classify each triangle as an equilateral triangle, an isosceles triangle, or a scalene triangle.

4 in.

5 in.

6 in.

56°

12. Find x.

20°

13.The measure of the vertex angle of an isosceles triangle is 12°. Find the measure of each base angle.

14.Refer to rectangle EFGH shown below.

a.	Find m(HG).		b.	Find m(FH).
	Find m( FGH).			Find m(			).
c.	Find m( FGH).		d.	Find m(		EH	).
	E		12		F
		6.5	x		5
			x
	H				G
	H				G

16.Find the sum of the measures of the angles of a decagon.

17.Find the perimeter of the ﬁgure shown below.

25 in.

36 in.

42 in.

37 in.

48in.

18.The perimeter of an equilateral triangle is 45.6 m. Find the length of each side.

19.Find the area of the shaded part of the ﬁgure shown below.

8 cm

16 cm

10 cm

25 cm

20. DECORATING A patio has the shape of a trapezoid, as shown on the right. If indoor/

outdoor carpeting sells for $18 a

27 ft

square yard installed, how much will it cost to carpet the patio?

836	Chapter 9 Test

21.How many square inches are in one square foot?

22.Find the area of the rectangle shown below in square inches.

23.Refer to the ﬁgure below, where O is the center of the circle.

a.	Name each chord.	R
a.	Name each chord.	S
b.	Name a diameter.	X
c.	Name each radius.	Y

24.Fill in the blank: If C is the circumference of a circle and D is the length of its diameter, then DC .

In Problems 25–27, when appropriate, give the exact answer and an approximation to the nearest tenth.

25.Find the circumference of a circle with a diameter of 21 feet.

26.Find the perimeter of the ﬁgure shown below. Assume that the arcs are semicircles.

20 ft

12 ft

20 ft

27.HISTORY Stonehenge is a prehistoric monument in England, believed to have been built by the Druids. The site, 30 meters in diameter, consists of a circular arrangement of stones, as shown below. What area does the monument cover?

28.See the ﬁgure below, where MNO RST. Name the six corresponding parts of the congruent triangles.

N S

29.Tell whether each pair of triangles are congruent. If they are, tell why.

5 yd	5 yd	5 yd	5 yd
5 yd		5 yd
b.
39°	53°	39°	53°
7 cm		7 cm
c.
62°		62°
57°	61°	57°	61°
d.
81°		81°

30. Refer to the ﬁgure below, in which ABC DEF.

a. Find m(DE).		b.	Find m( E).
C			F
6 in.	7 in.
60°	50°
A	B	E	D

8in.

31.Tell whether the triangles in each pair are similar.

a.	b.	29°
		29°

43° 43°

29°

32. Refer to the triangles below. The units are meters.

a. Find x.		b.	Find y.
C			F
			F
6	y		4	8
A	B	D	x	E
A	B		x
	9

33.SHADOWS If a tree casts a 7-foot shadow at the same time as a man 6 feet tall casts a 2-foot shadow, how tall is the tree?

34.Refer to the right triangle below. Find the missing side length. Approximate any exact answers that contain a square root to the nearest tenth.

a.Find c if a 10 cm and b 24 cm.

b.Find b if a 6 in. and c 8 in.

35.TELEVISIONS To the nearest tenth of an inch, what is the diagonal measurement of the television screen shown below?

d in.		19 in.

25 in.

36. How many cubic inches are there in 1 cubic foot?

Find the volume of each ﬁgure. Give the exact answer and an approximation to the nearest hundredth if an answer contains p.

37.

38.

8 m

6 m

10 m

6 m

	Chapter 9 Test	837
39.	40.
	27 in.
	20 in.
24 in.	Area:
	30 in.2

41.	42.	3 yd
27 ft		7 yd

20 ft

21 ft

29 ft

43.

44.

12 mi

4 in.

10 mi

10mi

45.FARMING A silo is used to store wheat and corn. Find the volume of the silo shown below. Give the exact answer and an approximation to the nearest cubic foot.

40 ft

30 ft

46.Give a real-life example in which the concept of perimeter is used. Do the same for area and for volume. Be sure to discuss the type of units used in each case.

<<< < Предыдущая 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 6566 / 6766 67 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
10.12.2018408.58 Кб6antikrizisnoe-upravlenie.doc
#
18.12.2018108.03 Кб6ARHITYeKTURA.doc
#
10.06.20158.32 Mб301aswath_damodaran-investment_philosophies_2012.pdf
#
10.06.201582.94 Кб7audit_risk.doc
#
10.06.20154.17 Mб4BallotingApplicationDocs-SIMSUsersGuide.pdf
#
10.06.201515.55 Mб147basic mathematics for college students.pdf
#
10.06.2015901.06 Кб40Bazy_dannykh_i_znanii_UP_SHirokov_L.A._2000.pdf
#
29.03.20162.45 Mб35Bernd_Fon_Vittenburg_-_Shakh_planete_Zemlya.pdf
#
10.06.201561.93 Кб44bezopasnost.docx
#
10.06.20152.89 Mб84bibliofond_ru_604241.docx
#
10.06.20152.1 Mб30blind_watchmaker.pdf