Page 1 of 7
11.3
Goal
Use properties of arcs of
circles.
Key Words
• minor arc
• major arc
• semicircle
• congruent circles
• congruent arcs
• arc length
Arcs and Central Angles
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 䉺C that lie in the
interior of aACB form a minor arc .
Points A, B, and all the points on 䉺C
that do not lie on AB
s form a major arc .
You name an arc by its endpoints.
Use one other point on a major arc
as part of its name to distinguish it
from the minor arc.
A
minor arc
AB
r
B
C
major arc
ADB
t
D
The measures of a minor arc and a major arc depend on the
central angle of the minor arc.
Student Help
LOOK BACK
For the definition of
a central angle, see
p. 454.
A
The measure of a minor arc is the
measure of its central angle.
mAB
r ⴝ 60ⴗ
B
60
The measure of a major arc is the
difference of 360 and the measure
of the related minor arc.
C
mADB
t ⴝ 360ⴗ ⴚ 60ⴗ
ⴝ 300ⴗ
D
A semicircle is an arc whose central angle measures 180.
A semicircle is named by three points. Its measure is 180.
1
EXAMPLE
Name and Find Measures of Arcs
Name the red arc and identify the type of arc. Then find its measure.
a.
G
L
b.
E
K 110ⴗ
40ⴗ
D
M
F
N
Solution
a. DF
s is a minor arc. Its measure is 40.
b. LMN
t is a major arc. Its measure is 360 110 250.
11.3
Arcs and Central Angles
601
Page 2 of 7
Visualize It!
POSTULATE 16
A
Arc Addition Postulate
Words
C
B
Arcs of a circle are
adjacent if they intersect
only at their endpoints.
AB
r and BC
r are adjacent.
A
The measure of an arc formed by
two adjacent arcs is the sum of the
measures of the two arcs.
C
mACB
t mAC
s mCB
s
Symbols
Find Measures of Arcs
2
EXAMPLE
B
Find the measure of GEF
t.
G
Solution
H
40ⴗ
R 80ⴗ
110ⴗ
mGEF
t mGH
t mHE
t mEF
s
40ⴗ 80ⴗ 110ⴗ
E
F
230
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.
Identify Congruent Arcs
3
EXAMPLE
Find the measures of the blue arcs. Are the arcs congruent?
a.
A
b.
B 45ⴗ
45ⴗ
X
Z
65ⴗ
D
Y W
C
Solution
a. Notice that AB
s and DC
s are in the same circle. Because
mAB
s mDC
s 45, AB
s c DC
s.
b. Notice that XY
s and ZW
s are not in the same circle or in congruent
s 65, XY
s ç ZW
s.
circles. Therefore, although mXY
s mZW
Identify Congruent Arcs
Find the measures of the arcs. Are the arcs congruent?
602
Chapter 11
Circles
D
C
1. BC
s and EF
s
2. BC
s and CD
s
3. CD
s and DE
s
4. BFE
t and CBF
t
B
72ⴗ
58ⴗ A
100ⴗ 58ⴗ
F
E
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Student Help
SKILLS REVIEW
Arc Length An arc length is a portion of the circumference of
a circle. You can write a proportion to find arc length.
To review finding
circumference of a
circle, see p. 674.
A
arc length
central angle
r
arc length of AB
r
mAr
B
2πr
360
C
B
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
A
P
r
r
mA
B
Arc length of AB
s p 2πr
B
360
4
EXAMPLE
Find Arc Lengths
Find the length of the red arc.
a.
b.
5 cm
c.
A
50ⴗ
B
7 cm
7 cm
50ⴗ
C
E
98ⴗ
F
D
Student Help
Solution
50ⴗ
360
a. Arc length of AB
s p 2π(5) ≈ 4.36 centimeters
STUDY TIP
You can substitute 3.14
as an approximation of
π or use a calculator.
50 ⴗ
360
b. Arc length of CD
s p 2π(7) ≈ 6.11 centimeters
98ⴗ
360
c. Arc length of EF
s p 2π(7) ≈ 11.97 centimeters
Find Arc Lengths
Find the length of the red arc. Round your answer to the nearest
hundredth.
B
5.
6.
7.
2 in.
C 120ⴗ
A
D
180ⴗ
N
E
4 ft
M
90ⴗ
6 cm
F
11.3
Arcs and Central Angles
603
Page 4 of 7
11.3 Exercises
Guided Practice
Vocabulary Check
1. In the diagram at the right, identify a major arc,
B
A
a minor arc, and a semicircle.
C
D
2. Draw a circle with a pair of congruent arcs.
3. What is the difference between arc measure and
arc length?
Skill Check
Find the measure in 䉺T.
P
4. mRS
r
5. mRPS
t
6. mPQR
t
7. mQS
r
8. mQSP
t
9. maQTR
T
P
40ⴗ
R
60ⴗ
S
120ⴗ
Find the blue arc length. Round your answer to the nearest
hundredth.
10. Length of AB
s
11. Length of DE
s
2 yd
40ⴗ
D
D
A
C
12. Length of FGH
t
G
6 cm
C 100ⴗ
B
F
F
5m C
140ⴗ
H
E
Practice and Applications
Extra Practice
Naming Arcs Name the blue minor arc and find its measure.
See p. 695.
13.
14.
15.
E
P
135ⴗ
C
130ⴗ
P
C
C
L
N
150ⴗ
D
Naming Arcs Name the blue major arc and find its measure.
Homework Help
Example 1:
Example 2:
Example 3:
Example 4:
16.
Exs. 13–39
Exs. 30–42
Exs. 43–46
Exs. 47–54
17.
B
W
G
A
75ⴗ
X
C
C 160ⴗ
Chapter 11
Circles
F
C
H
30ⴗ
D
Y
604
18.
Page 5 of 7
Types of Arcs Determine whether the arc is a minor arc, a major arc,
&* and QU
&** are diameters.
or a semicircle of 䉺R. PT
19. PQ
r
20. SU
s
21. PQT
t
22. QT
s
23. TUQ
t
24. TUP
t
25. QUT
t
26. PUQ
t
P
P
S
R
T
U
Finding the Central Angle Find the measure of aACB.
27.
28.
A
C
A
165ⴗ
B
C
B
&* and JL
&* are diameters. Find
Measuring Arcs and Central Angles KN
the measure.
30. mKL
r
31. mMN
t
32. mLNK
t
33. mMKN
t
34. mNJK
t
35. maMQL
36. mML
r
37. maJQN
38. mJM
r
39. mLN
s
J
N
55ⴗ
P
60ⴗ
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.
9
nha
oro
s
Azore
eN
d
do
nan
Fer
ab
dth
Go
Greenwich
M
os
co
He
lsin w
ki
Rome
12
8
3
9
11
4
7
5
P.M.
6
8
Tashkent
hi
Karac
lles
che
Sey
. . . it is 4 P.M.
in San Francisco
10
7
5
A.M.
6
4
Ma
nila
Bangk
ok
2
3
A
nc
ho
ra
Ho
g
n
o
lulu e
Anady
r
éa
Noum
ney
Syd o
ky
To
10
Wellington
When it is
9 A.M. in Tokyo . . .
Noon
12
1
11
Car
aca
s
Bosto
n
1
2
In Exs. 30–39, copy the
diagram and add
information to it as you
solve the exercises, as
shown on p. 588.
180ⴗ
A
New Orleans
r
Denve cisco
n
Fra
San
VISUAL STRATEGY
C
90ⴗ
B
Student Help
29.
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
Page 6 of 7
IStudent Help
Naming Congruent Arcs Are the blue arcs congruent? Explain.
ICLASSZONE.COM
43.
J
HOMEWORK HELP
Extra help with problem
solving in Exs. 47–52 is
at classzone.com
44.
P
A
L
6
K
R
5
D
P
C
B
45.
46.
Z
T
4
60ⴗ
W
4
60ⴗ Y
E
K
J
X
U
G
F
H
Finding Arc Length Find the length of AB
r. Round your answer to the
nearest hundredth.
47.
48.
49.
B
A
A
B
A
P
45ⴗ P
3 cm
120ⴗ
10 ft
60ⴗ
7 in.
P
B
50.
51.
P
30ⴗ A
4 cm B
52.
A
12 m P
75ⴗ
A
B
150ⴗ
6
in.
P
B
53.
You be the Judge A friend tells you two arcs from different
circles have the same arc length if their central angles are equal.
Is your friend correct? Explain your reasoning.
54. Challenge Engineers reduced the lean of the Leaning Tower of
Pisa. If they moved it back 0.46, what was the arc length of the
move? Round your answer to the nearest whole number.
5588 cm
606
Chapter 11
Circles
Page 7 of 7
Standardized Test
Practice
Mixed Review
s in 䉺P shown below?
55. Multiple Choice What is the length of AC
A

C

B

D

5.6 ft
19.5 ft
8 ft
A 40ⴗ
P
B
25.1 ft
D
Finding Leg Lengths Find the lengths of the legs of the triangle.
Round your answers to the nearest tenth. (Lesson 10.5)
56.
57.
4
x
x
x
48ⴗ
58.
9
34ⴗ
y
56ⴗ
y
Algebra Skills
C
16.8 ft
y
14
Simplifying Ratios Simplify the ratio. (Skills Review, p. 660)
2 km
400 km
3 yards
27 ft
5 ft
72 in.
59. 60. 4 ounces
8 po u nd s
61. 62. Quiz 1
Tell whether the given line, segment, or point is best described as a
chord, a secant, a tangent, a diameter, a radius, or a point of
tangency. (Lesson 11.1)
^&*(
1. AB
2. ^&
JH*(
&**
3. GE
&*
4. JH
&*
5. CE
6. D
J
G
H
C
B
E
D
A
&* and PR
&* are tangent to 䉺C. Find the value of x. (Lesson 11.2)
PQ
7.
8.
P
P
x
C
P
9. P
x
P
C
2x ⴚ 7
R
P
xⴚ1
C
15
3x ⴚ 8
R
R
Find the length of AB
s. Round your answer to the nearest hundredth.
(Lesson 11.3)
10.
A
70ⴗ
C 3 cm B
11.
12.
A
150ⴗ
B
7
m
C
C
10 ft
B
11.3
25ⴗ
A
Arcs and Central Angles
607