11.3 Arcs and Central Angles - Murrieta Valley Unified ...

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11.3 Arcs and Central Angles

Goal

Use properties of arcs of circles.

Key Words

? minor arc ? major arc ? semicircle ? congruent circles ? congruent arcs ? arc length

Student Help

LOOK BACK For the definition of a central angle, see p. 454.

Any two points A and B on a circle C determine a minor arc and a major arc (unless the points lie on a diameter).

If the measure of aACB is less than 180, then A, B, and all the points on C that lie in the interior of aACB form a minor arc .

Points A, B, and all the points on C that do not lie on AsB form a major arc .

You name an arc by its endpoints.

Use one other point on a major arc

D

as part of its name to distinguish it

from the minor arc.

minor arc A ArB

B

C

major arc At DB

The measures of a minor arc and a major arc depend on the

central angle of the minor arc.

The measure of a minor arc is the measure of its central angle.

A mArB 60

B 60

The measure of a major arc is the

difference of 360 and the measure of the related minor arc.

C D

mAt DB 360 60 300

A semicircle is an arc whose central angle measures 180. A semicircle is named by three points. Its measure is 180.

EXAMPLE 1 Name and Find Measures of Arcs

Name the red arc and identify the type of arc. Then find its measure.

a.

G

b.

L

E

K 110

40 DF

M N

Solution a. DsF is a minor arc. Its measure is 40. b. Lt MN is a major arc. Its measure is 360 110 250.

11.3 Arcs and Central Angles 601

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Visualize It!

A

C B

Arcs of a circle are adjacent if they intersect only at their endpoints. ArB and BrC are adjacent.

POSTULATE 16

Arc Addition Postulate

Words The measure of an arc formed by

A

two adjacent arcs is the sum of the

C

measures of the two arcs.

Symbols mAt CB mAsC mCsB

B

EXAMPLE 2 Find Measures of Arcs

Find the measure of Gt EF .

Solution

mGt EF mGtH mHtE mEsF

40 80 110

230

F

G H

40 R 80 110

E

Two circles are congruent circles if they have the same radius. Two arcs of the same circle or of congruent circles are congruent arcs if they have the same measure.

EXAMPLE 3 Identify Congruent Arcs

Find the measures of the blue arcs. Are the arcs congruent?

a. A B 45

D 45

C

b.

XZ

65

YW

Solution a. Notice that AsB and DsC are in the same circle. Because mAsB mDsC 45, AsB c DsC .

b. Notice that XsY and Zs W are not in the same circle or in congruent circles. Therefore, although mXsY mZs W 65, XsY ? Zs W .

Identify Congruent Arcs

Find the measures of the arcs. Are the arcs congruent?

1. BsC and EsF

2. BsC and CsD

C

D

B

58

72 A

100 58

E

3. CsD and DsE

4. Bt FE and Ct BF

F

602 Chapter 11 Circles

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Student Help

SKILLS REVIEW To review finding circumference of a circle, see p. 674.

Arc Length An arc length is a portion of the circumference of a circle. You can write a proportion to find arc length.

A

r C

B

arc length

central angle

arc len2g thr of ArB m36A r 0B

full circumference

full circle

ARC LENGTH

Words In a circle, the ratio of the length of a given

arc to the circumference is equal to the ratio of the measure of the arc to 360.

Symbols Arc length of AsB m36Ar 0B p 2r

A P

rB

Student Help

STUDY TIP You can substitute 3.14 as an approximation of or use a calculator.

EXAMPLE 4 Find Arc Lengths

Find the length of the red arc.

a.

b.

c.

5 cm A

50 B

C 7 cm 50

D

Solution a. Arc length of AsB 35600 p 2(5) 4.36 centimeters b. Arc length of CsD 35600 p 2(7) 6.11 centimeters c. Arc length of EsF 39680 p 2(7) 11.97 centimeters

E 7 cm

98

F

Find Arc Lengths

Find the length of the red arc. Round your answer to the nearest hundredth.

5.

B

2 in.

C 120

6.

D

180

E

4 ft

7.

N

M

90 6 cm

A

F

11.3 Arcs and Central Angles 603

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11.3 Exercises

Guided Practice

Vocabulary Check

1. In the diagram at the right, identify a major arc,

a minor arc, and a semicircle.

A

B

2. Draw a circle with a pair of congruent arcs.

C D

3. What is the difference between arc measure and arc length?

Skill Check

Find the measure in T.

4. mRrS

5. mRt PS

6. mPt QR

7. mQrS

8. mQt SP

9. maQTR

P 40

T

R

P

60

120

S

Find the blue arc length. Round your answer to the nearest hundredth.

10. Length of AsB

11. Length of DsE

12. Length of Ft GH

A 2 yd

D

C 40 B

D 6 cm C 100

F

E

G

5m C

F

140 H

Practice and Applications

Extra Practice

See p. 695.

Naming Arcs Name the blue minor arc and find its measure.

13. P 135 P C

14.

E

130 C D

15. L

C 150 N

Homework Help

Example 1: Exs. 13?39 Example 2: Exs. 30?42 Example 3: Exs. 43?46 Example 4: Exs. 47?54

604 Chapter 11 Circles

Naming Arcs Name the blue major arc and find its measure.

16.

B

17.

W

18.

A

75 C

D

X C 160 Y

FC

G

H 30

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Student Help

VISUAL STRATEGY In Exs. 30?39, copy the diagram and add information to it as you solve the exercises, as shown on p. 588.

Types of Arcs Determine whether the arc is a minor arc, a major arc,

or a semicircle of R. P&T* and Q&U** are diameters.

19. PrQ

20. SsU

P P

21. Pt QT 23. Tt UQ 25. Qt UT

22. QsT 24. Tt UP 26. Pt UQ

R

S

T U

Finding the Central Angle Find the measure of aACB.

27.

A

28.

29.

180

C

165

B

A

C

90 B

A

B

C

Measuring Arcs and Central Angles K&N* and J&L* are diameters. Find the measure.

30. mKrL

31. mMtN

J

32. mLt NK 34. mNt JK 36. mMrL 38. mJrM

33. mMt KN 35. maMQL 37. maJQN 39. mLsN

N

P

55 60

K

M L

Time Zone Wheel In Exercises 40?42, use the following information. To find the time in Tokyo when it is 4 P.M. in San Francisco, rotate the small wheel until 4 P.M. and San Francisco line up as shown. Then look at Tokyo to see that it is 9 A.M. there.

When it is 9 A.M. in Tokyo . . .

A.M.

5 67

8

Honolulu Anadyr Wellington Noum?a

Sydney

10

Ma9nTiolakyo Bangkok

Noon 11 12 1

Tashkent

Noronha

KarSaecyhciMheolslecosw

4

d e

2 3

Boston

New Orleans

DeSnavnerFArnacnhocirsagceo

8

5 4

7

6

P.M.

. . . it is 4 P.M. in San Francisco

Greenwich AzFoerrensando

Car9aGcodatshab

Helsinki Rome

3

10

2

11 12 1

Midnight

40. What is the arc measure for each time zone on the wheel?

41. What is the measure of the minor arc from the Tokyo zone to the Anchorage zone?

42. If two cities differ by 180 on the wheel, then it is 3:00 P.M. in one city when it is __?__ in the other city.

11.3 Arcs and Central Angles 605

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