10 6 standardized test prep circles and arcs

Lesson 10-6 Circles and Arcs 649

10-6

Objectives To fi nd the measures of central angles and arcs To fi nd the circumference and arc length

Circles and Arcs

Whoa—that wheel has to complete over a thousand rotations to go one mile.

The bicycle wheel shown at the right travels 63 in. in one complete rotation. If the wheel rotates only 1208 about the center, how far does it travel? Justify your reasoning.

In a plane, a circle is the set of all points equidistant from a given point called the center. You name a circle by its center. Circle P ((P) is shown below.

A diameter is a segment that contains the center of a circle and has both endpoints on the circle. A radius is a segment that has one endpoint at the center and the other endpoint on the circle. Congruent circles have congruent radii. A central angle is an angle whose vertex is the center of the circle.

P A B

C

P A B

C APC is a central

angle.

AB is a diameter.

PC is a radius.

P is the center of the circle.

Essential Understanding You can fi nd the length of part of a circle’s circumference by relating it to an angle in the circle.

An arc is a part of a circle. One type of arc, a semicircle, is half of a circle. A minor arc is smaller than a semicircle. A major arc is larger than a semicircle. You name a minor arc by its endpoints and a major arc or a semicircle by its endpoints and another point on the arc.

R

S P

TT

R

S

STR is a major arc.

RS is a minor arc.

Lesson Vocabulary

• circle • center • diameter • radius • congruent circles • central angle • semicircle • minor arc • major arc • adjacent arcs • circumference • pi • concentric circles • arc length • congruent arcs

L V L V

• c

LL VVV

• c

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Problem 1

Got It?

650 Chapter 10 Area

Naming Arcs

A What are the minor arcs of O?

Th e minor arcs are AD 0

, CE 0

, AC 0

, and DE 0

.

B What are the semicircles of O?

Th e semicircles are ACE 1

, CED 1

, EDA 1

, and DAC 1

.

C What are the major arcs of O that contain point A?

Th e major arcs that contain point A are ACD 1

, CEA 1

, EDC 1

, and DAE 1

.

1. a. What are the minor arcs of (A? b. What are the semicircles of (A? c. What are the major arcs of (A that contain point Q ?

Adjacent arcs are arcs of the same circle that have exactly one point in common. You can add the measures of adjacent arcs just as you can add the measures of adjacent angles.

A C

O ED

S P

A Q

R

Key Concept Arc Measure

Arc Measure Th e measure of a minor arc is equal to the measure of its corresponding central angle.

Th e measure of a major arc is the measure of the related minor arc subtracted from 360.

Th e measure of a semicircle is 180.

Example mRT 0

5 m/RST 5 50 mTQR 1

5 360 2 mRT 0

5 310S 50

R

T Q

Postulate 10-2 Arc Addition Postulate

Th e measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

mABC 1

5 mAB 0

1 mBC 0

B C

A

A

Th

Th

Th

How can you identify the minor arcs? Since a minor arc contains all the points in the interior of a central angle, start by identifying the central angles in the diagram.

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Problem 2

Got It?

Lesson 10-6 Circles and Arcs 651

Finding the Measures of Arcs

What is the measure of each arc in O?

A BC 0

mBC 0

5 m/BOC 5 32

B BD 0

mBD 0

5 mBC 0

1 mCD 0

mBD 0

5 32 1 58 5 90

C ABC 1

ABC 1

is a semicircle.

mABC 1

5 180

D AB 0

mAB 0

5 180 2 32 5 148

2. What is the measure of each arc in (C? a. mPR

0

b. mRS 0

c. mPRQ 1

d. mPQR 1

Th e circumference of a circle is the distance around the circle. Th e number pi (p) is the ratio of the circumference of a circle to its diameter.

Th e number p is irrational, so you cannot write it as a terminating or repeating decimal. To approximate p, you can use 3.14, 227 , or the key on your calculator.

Coplanar circles that have the same center are concentric circles.

Concentric circles

B

O

C D

A

32

58

P R

C

S Q

77

28

Theorem 10-9 Circumference of a Circle

Th e circumference of a circle is p times the diameter.

C 5 pd or C 5 2pr O

C

rd

W

A

B

C

How can you fi nd m BD 0

? BD 0

is formed by adjacent arcs BC

0

and CD 0

. Use the Arc Addition Postulate.

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Problem 3

Got It?

8 ft

Outer edge

Inner edge

2 ft

652 Chapter 10 Area

Finding a Distance

Film A 2-ft-wide circular track for a camera dolly is set up for a movie scene. Th e two rails of the track form concentric circles. Th e radius of the inner circle is 8 ft. How much farther does a wheel on the outer rail travel than a wheel on the inner rail of the track in one turn?

circumference of inner circle 5 2pr

5 2p(8) Substitute 8 for r.

5 16p Simplify.

Th e radius of the outer circle is the radius of the inner circle plus the width of the track.

radius of the outer circle 5 8 1 2 5 10

circumference of outer circle 5 2pr

5 2p(10) Substitute 10 for r.

5 20p Simplify.

Th e diff erence in the two distances traveled is 20p 2 16p, or 4p ft.

4p < 12.56637061 Use a calculator.

A wheel on the outer edge of the track travels about 13 ft farther than a wheel on the inner edge of the track.

3. a. A car has a circular turning radius of 16.1 ft. Th e distance between the two front tires is 4.7 ft. How much farther does a tire on the outside of the turn travel than a tire on the inside?

b. Reasoning Suppose the radius of (A is equal to the diameter of (B. What is the ratio of the circumference of (A to the circumference of (B? Explain.

Use the formula for the circumference of a circle.

Use the formula for the circumference of a circle.

16.1 ft

4.7 ft

F t H r

What do you need to fi nd? You need to fi nd the distance around the track, which is the circumference of a circle.

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Problem 4

Got It?

Lesson 10-6 Circles and Arcs 653

Th e measure of an arc is in degrees while the arc length is a fraction of a circle’s circumference. An arc of 608 represents 60360 or

1 6 of the circle. Its arc length is

1 6 the

circumference of the circle. Th is observation suggests the following theorem.

Finding Arc Length

What is the length of each arc shown in red? Leave your answer in terms of π.

A B

length of XY 0

5 mXY 0

360 ? pd length of XPY 1

5 mXPY 1

360 ? 2pr

5 90360 ? p(16) Substitute. 5 240 360 ? 2p(15)

5 4p in. Simplify. 5 20p cm

4. What is the length of a semicircle with radius 1.3 m? Leave your answer in terms of p.

It is possible for two arcs of diff erent circles to have the same measure but diff erent lengths. It is also possible for two arcs of diff erent circles to have the same length but diff erent measures. Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles.

Theorem 10-10 Arc Length

Th e length of an arc of a circle is the product of the ratio measure of the arc

360 and the circumference of the circle.

length of AB 0

5 mAB 0

360 ? 2pr

5 mAB 0

360

? pd

r A

B O

16 in.

X

O Y

15 cm

240

P

X

Y O

Use a formula for arc length.

X

Y O

60

XY RS

O P

R

S P

60

GG

How do you know which formula to use? It depends on whether the diameter is given or the radius is given.

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Lesson Check

654 Chapter 10 Area

Do you know HOW? Use P at the right to answer each question. For Exercises 5 and 6, leave your answers in terms of π.

1. What is the name of a minor arc?

2. What is the name of a major arc?

3. What is the name of a semicircle?

4. What is mAB 0

?

5. What is the circumference of (P?

6 What is the length of BD 0

?

Do you UNDERSTAND? 7. Vocabulary What is the diff erence between the

measure of an arc and arc length? Explain.

8. Error Analysis Your class must fi nd the length of AB

0

. A classmate submits the following solution. What is the error?

A B

C D

P

81

65

9 cm

O A

B

C

70

4 m

Length of AB = · 2 r

· 2 (4)

mAB 360 110 360

=

=

22 9 m

Practice and Problem-Solving Exercises

Name the following in O.

9. the minor arcs

10. the major arcs

11. the semicircles

Find the measure of each arc in P.

12. TC 0

13. TBD 1

14. BTC 1

15. TCB 1

16. CD 0

17. CBD 1

18. TCD 1

19. DB 0

20. TDC 1

21. TB 0

22. BC 0

23. BCD 1

Find the circumference of each circle. Leave your answer in terms of π.

24. 25. 26. 27.

28. Th e camera dolly track in Problem 3 can be expanded so that the diameter of the outer circle is 70 ft. How much farther will a wheel on the outer rail travel during one turn around the track than a wheel on the inner rail?

PracticeA See Problem 1.F

E D

C

B O

See Problem 2.

B

C T

DP

128

See Problem 3.

20 cm

3 ft 4.2 m

14 in.

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Lesson 10-6 Circles and Arcs 655

29. Th e wheel of a compact car has a 25-in. diameter. Th e wheel of a pickup truck has a 31-in. diameter. To the nearest inch, how much farther does the pickup truck wheel travel in one revolution than the compact car wheel?

Find the length of each arc shown in red. Leave your answer in terms of π.

30. 31. 32.

33. 34. 35.

36. Think About a Plan Nina designed a semicircular arch made of wrought iron for the top of a mall entrance. Th e nine segments between the two concentric semicircles are each 3 ft long. What is the total length of wrought iron used to make this structure? Round your answer to the nearest foot.

• What do you know from the diagram? • What formula should you use to fi nd the amount of

wrought iron used in the semicircular arches?

Find each indicated measure for O.

37. m/EOF 38. mEJH 1

39. mFH 0

40. m/FOG 41. mJEG 1

42. mHFJ 1

43. Pets A hamster wheel has a 7-in. diameter. How many feet will a hamster travel in 100 revolutions of the wheel?

44. Traffi c Five streets come together at a traffi c circle, as shown at the right. Th e diameter of the circle traveled by a car is 200 ft. If traffi c travels counterclockwise, what is the approximate distance from East St. to Neponset St.?

227 ft 454 ft

244 ft 488 ft

45. Writing Describe two ways to fi nd the arc length of a major arc if you are given the measure of the corresponding minor arc and the radius of the circle.

See Problem 4.

14 cm 45

24 ft

60 18 m

36 in.

30 23 m 9 m 25

ApplyB

20 ft

O E

F G

H

J

70

40 East St.

Rt e.

1

Neponset St.

Ma in

St . Maple St.

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656 Chapter 10 Area

46. Time Hands of a clock suggest an angle whose measure is continually changing. How many degrees does a minute hand move through during each time interval?

a. 1 min b. 5 min c. 20 min

Algebra Find the value of each variable.

47. 48.

49. Landscape Design A landscape architect is constructing a curved path through a rectangular yard. Th e curved path consists of two 908 arcs. He plans to edge the two sides of the path with plastic edging. What is the total length of plastic edging he will need? Round your answer to the nearest meter.

50. Reasoning Suppose the radius of a circle is doubled. How does this aff ect the circumference of the circle? Explain.

51. A 608 arc of (A has the same length as a 458 arc of (B. What is the ratio of the radius of (A to the radius of (B?

Find the length of each arc shown in red. Leave your answer in terms of π.

52. 53. 54.

55. Coordinate Geometry Find the length of a semicircle with endpoints (1, 3) and (4, 7). Round your answer to the nearest tenth.

56. In (O, the length of AB 0

is 6p cm and mAB 0

is 120. What is the diameter of (O?

57. Th e diagram below shows two 58. Given: (P with AB 6 PC concentric circles. AR > RW. Prove: mBC

0

5 mCD 0

Show that the length of ST 0

is equal to the length of QR

0

.

AP

Q c (4c – 10)

R

Q

P(x 40)

(2x 60)

(3x 20)

A

2 m

2 m

4 m

4 m

45

4.1 ft 50

7.2 in. 6 m

ChallengeC Proof

U

V Q

A

T

R W

S

B C

D P

A

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Lesson 10-6 Circles and Arcs 657

59. Sports An athletic fi eld is a 100 yd-by-40 yd rectangle, with a semicircle at each of the short sides. A running track 10 yd wide surrounds the fi eld. If the track is divided into eight lanes of equal width, what is the distance around the track along the inside edge of each lane?

100 yd 10 yd

40 yd

Mixed Review

Part of a regular dodecagon is shown at the right.

64. What is the measure of each numbered angle?

65. Th e radius is 19.3 mm. What is the apothem?

66. What is the perimeter and area of the dodecagon to the nearest millimeter or square millimeter?

Can you conclude that the fi gure is a parallelogram? Explain.

67. 68. 69.

Get Ready! To prepare for Lesson 10-7, do Exercises 70 and 71.

70. What is the circumference of a circle with diameter 17 in.?

71. What is the length of a 908 arc in a circle with radius 6 cm?

See Lesson 10-5.

O

2

3 4

1

a r

See Lesson 6-3.

See Lesson 10-6.

Standardized Test Prep

60. Th e radius of a circle is 12 cm. What is the length of a 608 arc?

3p cm 4p cm 5p cm 6p cm

61. What is the image of P for a 1358 clockwise rotation about the center of the regular octagon?

S U

T R

62. Which of the following are the sides of a right triangle?

6, 8, 12 8, 15, 17 9, 11, 23 5, 12, 15

63. Quadrilateral ABCD has vertices A(1, 1), B(4, 1), C(4, 6), and D(1, 6). Quadrilateral RSTV has vertices R(23, 4), S(23, 22), T(213 22), and V(213, 4). Show that ABCD and RSTV are similar rectangles.

SAT/ACT

P Q

R

S

TU V

W

Extended Response