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Chapter 5 l Skills Practice 399
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Skills Practice Skills Practice for Lesson 5.1
Name _____________________________________________ Date ____________________
Riding a Ferris WheelIntroduction to Circles
Vocabulary Match each definition to its corresponding term.
1. the set of all points equidistant from a point a. arc
c. circle
2. the distance from a point on a circle to the center b. central angle
i. radius
3. a line segment whose endpoints lie on a circle c. circle
d. chord
4. a chord that passes through the center of a circle d. chord
e. diameter
5. a line that intersects a circle at exactly two points e. diameter
j. secant
6. a line that intersects a circle at exactly one point f. inscribed angle
l. tangent
7. an angle whose vertex is the center of a circle g. major arc
b. central angle
8. an angle whose vertex lies on the circle and whose h. minor arc
sides are chords of the circle
f. inscribed angle
9. an unbroken portion of a circle that lies between i. radius
two points on the circle
a. arc
10. an arc whose endpoints lie on the diameter j. secant
k. semicircle
11. an arc that is less than a semicircle k. semicircle
h. minor arc
12. an arc that is greater than a semicircle l. tangent
g. major arc
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400 Chapter 5 l Skills Practice
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Problem Set Use the given circle to answer each question.
1. a. Name the circle.
The name of the circle is circle T.
b. Name each point shown on the circle.
The points on the circle are points H, J, K, and L.
c. Name the point at the center of the circle.
The point at the center of the circle is point T.
2. a. Name the circle.
The name of the circle is circle X.
b. Name each point shown on the circle.
The points on the circle are points A, B, and C.
c. Name the point at the center of the circle.
The point at the center of the circle is point X.
3. Name a radius of circle P.
Segment PX is a radius of the circle (also segments PW and PY).
4. Name a diameter of circle P.
A diameter of circle P is chord WY.
5. Name a chord on circle P that is not a diameter.
Chord VX is not a diameter.
6. Name a radius of circle A.
Segment AC is a radius of the circle (also segments AB and AD).
H
TJ
LK
B C
X
A
P
Y
W X
V
A
D
C
E
B
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7. Name a diameter of circle A.
A diameter of circle A is chord BD.
8. Name a chord on circle A that is not a diameter.
Chord BE is not a diameter.
9. Use a straightedge to draw each of the following on circle O.
a. chord BG
b. secant BN
c. tangent MN
Sample answers
10. Use a straightedge to draw each of the following on circle Y.
a. chord LM
b. secant LK
c. tangent LP
Sample answers
O
B G
M
N
YL
P
M
K
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402 Chapter 5 l Skills Practice
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Determine whether each angle is an inscribed angle, a central angle, or neither.
M
L
J
S
N
K
P
11. �MSN 12. �MLK
central angle inscribed angle
13. �KJM 14. �NSL
neither central angle
15. �KMN 16. �KPN
inscribed angle neither
Determine whether each arc is a semicircle, a minor arc, or a major arc.
R
A B
D
C
17. ADC 18. DA
semicircle minor arc
19. BDC 20. CBA
major arc semicircle
21. BC 22. ABD
minor arc major arc
�
�
�
�
�
�
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Skills Practice Skills Practice for Lesson 5.2
Name _____________________________________________ Date ____________________
Holding the WheelCentral Angles, Inscribed Angles, and Intercepted Arcs
Vocabulary Use the diagram of circle P to answer Questions 1 through 4.
P
A B
D C
1. Name all of the central angles in the diagram.
� APB, � BPC, � CPD, � DPA
2. Name all of the inscribed angles in the diagram.
� ACB, � CBD
3. Name all of the minor arcs in the diagram.
AB, BC, CD, DA
4. Name all of the intercepted arcs in the diagram.
AB, CD
5. Describe how to calculate the measure of a minor arc if you know the measure of
its central angle.
The measure of a minor arc is equal to the measure of its central angle.
6. Describe how to calculate the measure of an inscribed angle if you know the
measure of its central angle.
The measure of an inscribed angle is equal to half the measure of its central angle.
� � � �
� �
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404 Chapter 5 l Skills Practice
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Problem Set Use the given information to determine the measure of the indicated arc.
1. m�ATB � 36° 2. m�CMD � 113°
mAB � mCD �
A
B
T
C D
M
3. m�FZG � 90° 4. m�JVK � 59°
mFG � mJK �
F G
Z
J
KV
Use the given information to determine the measure of the indicated angle.
5. mBC � 46° 6. mUV � 19.5°
m�BAC � m�UWV �
B
C
A
U V
W
7. mMN � 140.5° 8. mHK � 161°
m�MQN � m�HNK �
M N
Q
K
H
N
36°
90° 59°
113°
46°
140.5°
19.5°
161°
� �
� �
� �
� �
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Use the given information to determine the measure of the indicated arc.
9. m�DEF � 84° 10. m�PQR � 31°
mDF � mPR �
D
F
EO
P R
Q
A
11. m�WXY � 19° 12. m�JKL � 55.5°
mWY � m JL �
X
W
Y
P
L
J
K
C
Use the given information to determine the measure of the indicated angle.
13. mYZ � 28° 14. mLK � 87°
m�YXZ � m�LMK �
Y
Z
XO
L K
M
N
15. mQR � 165° 16. mST � 102°
m�QPR � m�SWT �
P
Q
R
T
WS
T
A
168° 62°
38° 111°
14° 43.5°
82.5° 51°
� �
� �
� �
� �
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Skills Practice Skills Practice for Lesson 5.3
Name _____________________________________________ Date ____________________
Manhole CoversMeasuring Angles Inside and Outside of Circles
Vocabulary Match each diagram to the term that best describes it.
1. 2.
c. diameter d. inscribed angle
3. 4.
a. central angle b. chord
5. 6.
f. tangent e. secant
a. central angle
b. chord
c. diameter
d. inscribed angle
e. secant
f. tangent
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408 Chapter 5 l Skills Practice
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Problem Set Use the given information to determine the measure of the indicated central or inscribed angle.
1. 2.
A
BX
C
D
K
mAB � 40º mCD � 120º
m�AXB � m�CKD �
3. 4.
A
BL
CD
N
mAB � 40º mCD � 120º
m�ALB � m�CND �
40° 120°
20° 60°
� �
� �
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Use the given arc measures to determine the measure of the indicated
angle.
6.
mPS � 170º
mQR � 28º
m�PXS �
m�SXR �
m�PXS � 1 __ 2 (mPS � mQR)
m�PXS � 1 __ 2 (170º � 28º)
m�PXS � 1 __ 2 (198º)
m�PXS � 99º
m�SXR � 180º � 99º
� 81º
R
Q
P
S
X
5.
N M
L
O R
mLM � 90º
mON � 36º
m�LRM �
m�NRM �
m�LRM � 1 __ 2 (mLM � mON)
m�LRM � 1 __ 2 (90º � 36º)
m�LRM � 1 __ 2 (126º)
m�LRM � 63º
m�NRM � 180º � 63º
� 117º
��
��
� �� �
63°
117°
99°
81°
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7.
mED � 140º
mCD � 10º
m�EAD �
m�CAD �
Because arc BCD is a semicircle, its
measure is 180º.
mBCD � mBC � mCD
180º � mBC � 10º
mBC � 170º
m�EAD � 1 __ 2 (mED � mBC)
m�EAD � 1 __ 2 (140º � 170º)
m�EAD � 1 __ 2 (310º)
m�EAD � 155º
m�CAD � 180º � 155º
� 25º
E
D
CBA
8.
mXY � 20º
mYZ � 50º
m�XVY �
m�YVZ �
Because arc YZW is a semicircle, its
measure is 180º.
mYZW � m�YZ � m�ZW
180º � 50º � mZW
mZW � 130º
m�XVY � 1 __ 2 (mXY � mZW )
m�XVY � 1 __ 2 (20º � 130º)
m�XVY � 1 __ 2 (150º)
m�XVY � 75º
m�YVZ � 180º � 75º
� 105º
W
Z
YX
V
155°
75°25°
105°
�
�
� �
�
�
�
� �
�
�
� � �
�
�
��
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9. 10.
mBE � 20º mJK � 164º
mCD � 70º m I L � 42º
m�A � m�H �
m�A � 1 __ 2 (mCD � mBE ) m�H � 1 __
2 (mJK � m I L )
m�A � 1 __ 2 (70º � 20º) m�H � 1 __
2 (164º � 42º)
m�A � 1 __ 2 (50º) m�H � 1 __
2 (122º)
m�A � 25º m�H � 61º
11. 12.
Q OP
M
N
mAE � 170º mMQ � 50º
mBD � 20º mNP � 12º
m�C � m�O �
m�C � 1 __ 2 (mAE � mBD) m�O � 1 __
2 (mMQ � mNP)
m�C � 1 __ 2 (170º � 20º) m�O � 1 __
2 (50º � 12º)
m�C � 1 __ 2 (150º) m�O � 1 __
2 (38º)
m�C � 75º m�O � 19º
D
CBA
E
K
L
H
I
J
E
C
D
A B
25°
75°
61°
19°
� � � �
� � � �
�
�
�
�
�
�
�
�
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13. 14.
mLB � 108º mXP � 120º
m�LBC � m�XPO �
m�LBC � 1 __ 2 (mLB) m�XPO � 1 __
2 (mXP)
m�LBC � 1 __ 2 (108º) m�XPO � 1 __
2 (120º)
m�LBC � 54º m�XPO � 60º
Use the given angle measures to determine the degree measure of the
indicated arc.
15. 16.
m�AGF � 70º m�ABF � 20º
mAG � mBF �
m�AGF � 1 __ 2 (mAG) m�ABF � 1 __
2 (mBF )
70º � 1 __ 2 (mAG) 20º � 1 __
2 (mBF )
140º � mAG 40º � mBF
L
B CA
X
P QO
A
G
H
F
A
B
C
F
54° 60°
140° 40°
� �
� �
�
�
�
�
� �
� �
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17. 18.
mNQ � 160º mWY � 10º
m�L � 40º m�V � 10º
mMQ � mXY �
m�L � 1 __ 2 (mNQ � mMQ) m�V � 1 __
2 (mXY � mWY )
40º � 1 __ 2 (160º � mMQ) 10º � 1 __
2 (mXY � 10º)
80º � 160º � mMQ 20º � mXY � 10º
�80º � �mMQ 30º � mXY
mMQ � 80º
L
M
N
PQ
V
WX
ZY
80° 30°
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
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Use the given arc measures to determine the measure of the indicated angle.
19. 20.
mBCD � 228º mGFI � 254º
mBD � 132º mG I � 106º
m�A � m�H �
m�A � 1 __ 2 (mBCD � mBD) m�H � 1 __
2 (mGFI � mG I )
m�A � 1 __ 2 (228º � 132º) m�H � 1 __
2 (254º � 106º)
m�A � 1 __ 2 (96º) m�H � 1 __
2 (148º)
m�A � 48º m�H � 74º
A
D
B
C
H
F
G
I
48° 74°
� �
� �
� � ��
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21. 22.
mPSR � 235º mJML � 309º
mPR � 125º mJL � 51º
m�Q � m�K �
m�Q � 1 __ 2 (mPSR � mPR) m�K � 1 __
2 (mJML � mJL )
m�Q � 1 __ 2 (235º � 125º) m�K � 1 __
2 (309º � 51º)
m�Q � 1 __ 2 (110º) m�K � 1 __
2 (258º)
m�Q � 55º m�K � 129º
P
Q
R
S
J
M
L
K
55° 129°
� � � �
� �
� �
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Skills Practice Skills Practice for Lesson 5.4
Name _____________________________________________ Date ____________________
Color TheoryChords and Circles
Vocabulary 1. Describe how to use two chords to find the center of a circle.
To use two chords to find the center of a circle, find the perpendicular bisector of each chord. The intersection of the perpendicular bisectors is the center of the circle.
2. Can the perpendicular bisectors of parallel chords be used to find the center of a
circle? Why or why not?
The perpendicular bisectors of parallel chords cannot be used to find the center of a circle. The perpendicular bisectors of the chords are the same line or line segment.
3. Describe how the minor arcs of congruent chords are related.
The minor arcs of congruent chords are congruent. If two minor arcs in a circle are congruent, then their corresponding chords are congruent.
4. Define perpendicular bisector in your own words.
A perpendicular bisector is a line or line segment that is perpendicular to a segment at its midpoint.
Problem Set Use the diagram and your understanding of perpendicular bisectors to complete each statement.
D
B
X
C
A
H
G
M
R
S
1. ___
BA � ___
DA 3. SH � RH
2. BC � DC 4. ____
RM � ____
SM �
�
�
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Use a compass and straightedge to draw the perpendicular bisector of each chord.
5. 6.
7. 8.
Use the diagram and your understanding of congruency to complete each statement.
L
XM
NO
B
A
Z
X
Y
W
U
VS
9. ___
AX � ___
BX 10. ___
US � ___
VS
LO � MN XZ � YW
___
LO � ____
MN ___
XZ � ____
YW
___
LA � ____
AO � ____
MB � ____
BN ___
XU � ___
ZU � ___
YV � ____
WV
� �� �
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R
CD
P
Q
SB
K
F
G
JM
LN
11. ___
CB � ___
DB 12. ___
LM � ____
NM
PR � QS GF � KJ
___
PR � ___
QS ___
GF � ___
KJ
___
PC � ____
RC � ____
QD � ____
SD ___
GL � ___
FL � ____
KN � ___
JN
Locate the center of each circle using the given chords.
13. 14.
15. 16.
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Skills Practice Skills Practice for Lesson 5.5
Name _____________________________________________ Date ____________________
Solar EclipsesTangents and Circles
Vocabulary 1. Describe how the three terms tangent line, point of tangency, and tangent segment
are related. Identify similarities and differences.
A tangent line is a line that intersects a circle at exactly one point. That point is called the point of tangency. A tangent segment is a segment of a tangent line. One endpoint of the tangent segment is the point of tangency. All three terms involve one point on a circle. The point of tangency is one point that is on the circle.
2. Describe how the terms tangent and radius are related. Identify similarities and
differences.
Both the terms tangent and radius are related to circles. A circle has an infinite number of tangents and radii. A tangent intersects a circle at exactly one point. The remaining points on a tangent are on the exterior of a circle. A radius intersects a circle at exactly one point. The remaining points on a radius are on the interior of the circle, and end at the center. A tangent to a circle is perpendicular to the radius that is drawn from the point of tangency.
Problem Set Draw the indicated segment or line for each circle.
1. tangent ___
AD 2. tangent ___
XZ
A
D
B
O
Z
X
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3. radius ___
DK 4. radius ___
PT
K
D
T
P
Use a straight edge to draw a congruent tangent segment for each given tangent segment.
5. 6.
7. 8.
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5
Use the given measurements to determine the measure of each indicated angle.
9. 10.
X
R
T
S
A
C
D
B
m�RXS � 34º m�ACB � 70º
m�XRS � 73º m�BAC � 55º
m�RSX � 73º m�ABC � 55º
2(m�XRS) � m�RXS � 180º 2(m�ABC) � m�ACB � 180º
2(m�XRS) � 34º � 180º 2(m�ABC) � 70º � 180º
2(m�XRS) � 146º 2(m�ABC) � 110º
m�XRS � 73º m�ABC � 55º
m�XRS � m�RSX m�BAC � m�ABC
11. 12.
M
O
Y
N
J
K
X
L
m�OMN � 42º m�LJK � 80º
m�ONM � 42º m�JKL � 80º
m�MON � 96º m�JLK � 20º
m�OMN � m�ONM � 42º m�LJK � m�JKL � 80º
m�OMN � m�ONM � m�MON � 180º m�LJK � m�JKL � m�JLK � 180º
42º � 42º � m�MON � 180º 80º � 80º � m�JLK � 180º
84º � m�MON � 180º 160º � m�JLK � 180º
m�MON � 96º m�JLK � 20º
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13. Line AB is tangent to circle O.
m�AOC � 120º
m�BAO � 90º
m�ABO � 30º
m�AOB � 60º
m�BAO � 90º
m�BAO � m�ABO � 120º
90º � m�ABO � 120º
m�ABO � 30º
m�AOB � m�AOC � 180º
m�AOB � 120º � 180º
m�AOB � 180º � 120º
m�AOB � 60º
14. Line RU is tangent to circle S.
m�QRS � 142º
m�SUR � 90º
m�SRU � 38º
m�RSU � 52º
m�SUR � 90º
m�SRU � m�QRS � 180º
m�SRU � 142º � 180º
m�SRU � 38º
m�RSU � 180º � m�SUR � m�SRU
m�RSU � 180º � 90º � 38º
m�RSU � 52º
A
C
O
B
S
R
U
Q
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Skills Practice Skills Practice for Lesson 5.6
Name _____________________________________________ Date ____________________
GearsArc Length
Vocabulary
Define each term in your own words.
1. arc
An arc is an unbroken portion of a circle that lies between two points on the circle.
2. measure of a minor arc
The measure of a minor arc is the degree measure of its central angle.
3. arc length
Arc length is the measure of the length in linear units, such as inches or centimeters. It is a portion of the circumference.
Problem Set
Determine the measure of each minor arc.
1. L
M
T40°
4 in.
2.
A
BZ
100°
10 m
mLM � 40º mAB � 100º� �
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3.
S T
O
36°
8 cm
4.
D E
F
140°15 in.
mST � 36º mED � 140º
5. M
N
O120°
3 m
6. J
K
R
20 m20°
mMN � 120º mJK � 20º
7. B
C
A
5 cm
80°
8.
B
C
2 ftA
50°
mBC � 80º mBC � 50º
Calculate the circumference for each circle. Write your answers in terms of �.
9.
50 cm
10.
19 in.
C � 2�r C � 2�r C � 2�(50) C � 2�(19) C � 100� centimeters C � 38� inches
� �
� �
� �
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11.
10.5 ft
12.
4.2 m
C � 2�r C � 2�r C � 2�(10.5) C � 2�(4.2) C � 21� feet C � 8.4� meters
Calculate the arc length of the minor arc in each circle. Write your answers in terms of �.
13. L
M
T40°
4 in.
14.
A
BZ
100°
10 m
Arc length of LM: Arc length of AB:
( mLM _____ 360°
) 2�r � ( 40° _____ 360°
) 2�(4) ( mAB _____ 360°
) 2�r � ( 100° _____ 360°
) 2�(10)
� ( 1 __ 9 ) 8� � ( 5 ___
18 ) 20�
� 8 __ 9 � inches � 50 ___
9 � meters
15.
S T
O
36°
8 cm
16.
D E
F
140°15 in.
Arc length of ST: Arc length of DE:
( mST _____ 360°
) 2�r � ( 36° _____ 360°
) 2�(8) ( mDE _____ 360°
) 2�r � ( 140° _____ 360°
) 2�(15)
� ( 1 ___ 10
) 16� � ( 7 ___ 18
) 30� � 210 ____ 18
�
� 8 __ 5 � centimeters � 35 ___
3 � inches
� �
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17. M
N
O120°
3 m
18. J
K
R
20 m20°
Arc length of MN: Arc length of JK:
( mMN ______ 360°
) 2�r � ( 120° _____ 360°
) 2�(3) ( mJK _____ 360°
) 2�r � ( 20° _____ 360°
) 2�(20)
� ( 1 __ 3 ) 6� � ( 1 ___
18 ) 40�
� 2� meters � 20 ___ 9 � meters
19. B
C
A
5 cm
80°
20.
B
C
2 ftA
50°
Arc length of BC: Arc length of BC:
( mBC _____ 360°
) 2�r � ( 80° _____ 360°
) 2�(5) ( mBC _____ 360°
) 2�r � ( 50° _____ 360°
) 2�(2)
� ( 2 __ 9 ) 10� � ( 5 ___
36 ) 4�
� 20 ___ 9 � centimeters � 5 __
9 � feet
� �
� �
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Skills Practice Skills Practice for Lesson 5.7
Name _____________________________________________ Date ____________________
Playing DartsAreas of Parts of Circles
Vocabulary
Provide an example of each of the following. Use words and diagrams as necessary.
1. concentric circles 2. sector of circle
B
C
A
sector of circle
The two circles have the same center.
3. segment of circle
B
C
A
segment ofcircle
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Problem Set
List the names of the radii and the arc that is intercepted by the radii that form each sector.
1.
P
W X 2.
AD
C
radius: ____
PW radius: ___
AC radius:
___ PX radius:
___ AD
arc: WX arc: CD
3.
G
J
K
4.
Q
R
Z
radius: ____
GJ radius: ___
QR radius:
___ GK radius:
___ QZ
arc: JK arc: RZ
Calculate the area of each circle. Use 3.14 for �.
5.
T
4 in.
6.
A
L
9 cm
A � �r2 A � �r2
A � �(4)2 A � �(9)2
A � 3.14(16) A � 3.14(81)A � 50.24 square inches A � 254.34 square centimeters
� �
� �
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5
7.
6.5 mm
8.
18.4 ft
A � �r2 A � �r2
A � �(6.5)2 A � �(18.4)2
A � 3.14(42.25) A � 3.14(338.56)A � 132.665 square millimeters A � 1063.0784 square feet
Calculate the area of each sector. Use 3.14 for �. Round to the nearest hundredth, if necessary.
9.
A
B
O
40°3 cm
10. D
E
F120°
2 in.
A � 40 ____ 360
�r2 A � 120 ____ 360
�r2
A � 1 __ 9 �(3)2 A � 1 __
3 �(2)2
A � 1 __ 9 �(9) A � 1 __
3 (3.14)(4)
A � � A � 4.19 square inches
A � 3.14 square centimeters
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11.
S
R
160°
0.5 m
12. A
CB
10 cm
20°
A � 160 ____ 360
�r2 A � 20 ____ 360
�r2
A � 4 __ 9 �(0.5)2 A � 1 ___
18 �(10)2
A � 4 __ 9 (3.14)(0.25) A � 1 ___
18 (3.14)(100)
A � 0.35 square meters A � 17.44 square centimeters
13. B
C
A
6 cm
80°
14.
D
F
2 ft
E
50°
A � 80 ____ 360
�r2 A � 50 ____ 360
�r2
A � 2 __ 9 �(6)2 A � 5 ___
36 �(2)2
A � 2 __ 9 (3.14)(36) A � 5 ___
36 (3.14)(4)
A � 25.12 square centimeters A � 1.74 square feet
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Name _____________________________________________ Date ____________________
5
Calculate the area of the indicated triangle. Round to the nearest hundredth, if necessary.
15. B
C
A
3 in
Area of �BAC � 4.5 square inches
Area of �BAC � 1 __ 2 bh
� 1 __ 2 (3)(3)
� 9 __ 2 � 4.5 square inches
16.
L
J
K
16 m
Area of �JKL � 128 square meters
Area of �JKL � 1 __ 2 bh
� 1 __ 2 (16)(16)
� 256 ____ 2 � 128 square meters
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17. X
Y
Z
2 cm
60°
Area of �XYZ � √__
3 � 1.73 square centimeters
�XYZ is an equilateral triangle, so the base is 2 centimetres, and the height is √
__ 3 centimeters.
Area of �XYZ � 1 __ 2 bh
� 1 __ 2 (2)( √
__ 3 )
� √__
3 � 1.73 square centimeters
18.
5 ftQ
60° R
S
Area of �QRS � 6.25 √__
3 � 10.83 square feet
�QRS is an equilateral triangle, so the base is 5 feet, and the height is 2.5 √__
3 feet.
Area of �QRS � 1 __ 2 bh
� 1 __ 2 (5)(2.5 √
__ 3 )
� 6.25 √__
3 � 10.83 square feet
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5
Calculate the area of the shaded segment of the circle.
19. B
C
A
3 in.
The area of the shaded segment � the area of sector ABC � the area of triangle ABC.
Area of sector ABC � 90 ____ 360
�r2
� 1 __ 4 �(3)2
� 1 __ 4 (3.14)(9)
� 7.07 square inches
Area of �ABC � 1 __ 2 bh
� 1 __ 2 (3)(3)
� 9 __ 2 � 4.5 square inches
Area of segment � 7.07 � 4.5 � 2.57 square inches
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20.
M
C
O12 in.
The area of the shaded segment � the area of sector MOC � the area of triangle MOC.
Area of sector MOC � 90 ____ 360
�r2
� 1 __ 4 �(12)2
� 1 __ 4 (3.14)(144)
� 113.04 square inches
Area of �MOC � 1 __ 2 bh
� 1 __ 2 (12)(12)
� 72 square inches
Area of segment � 113.04 � 72 � 41.04 square inches
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21.
X
Y
Z
2 cm
60°
The area of the shaded segment � the area of sector XYZ � the area of triangle XYZ.
Area of sector XYZ � �r2
� 60 ____ 360
�(2)2
� 1 __ 6 (3.14)(4)
� 2.09 square centimeters
�XYZ is an equilateral triangle, so the base is 2 centimeters and the height is √
__ 3 centimeters.
Area of �XYZ � 1 __ 2 bh
� 1 __ 2 (2)( √
__ 3 )
� √__
3 � 1.73 square centimeters
Area of segment � 2.09 � 1.73 � 0.36 square centimeters
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22.
RP
Q
6 cm 60°
The area of the shaded segment � the area of sector PQR � the area of triangle PQR.
Area of sector PQR � 60 ____ 360
�r2
� 60 ____ 360
�(6)2
� 1 __ 6 (3.14)(36)
� 18.84 square centimeters
�PQR is an equilateral triangle, so the base is 6 centimeters and the height is 3 √
__ 3 centimeters.
Area of �XYZ � 1 __ 2 bh
� 1 __ 2 (6)(3 √
__ 3 )
� 15.59 square centimeters
Area of segment � 18.84 � 15.59 � 3.25 square centimeters
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Skills Practice Skills Practice for Lesson 5.8
Name _____________________________________________ Date ____________________
Crop CirclesCircle Measurements and Relationships
Vocabulary Write the term that best completes each statement.
1. Circles that share the same center but have different radii
lengths are called concentric.
2. Arc length is measured in linear units.
3. When an arc measure is 180º, the arc is called a semicircle .
4. A central angle is equal to the measure of its intercepted arc.
5. A circle segment is bounded by an arc and the line segment that intercepts
the arc endpoints.
6. A tangent to a circle is perpendicular to the radius that is drawn from the point
of tangency.
Problem Set Use the given information to determine the measure of the indicated central or inscribed angle.
1.
A
B
C
2.
L
R
N
mAB = 60º mLR = 110º
m�ACB = 60º m�LNR = 110º
� �
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3. L
N
M
4.
X
Z
Y
mLN = 48º mXZ = 170º
m�LMN = 24º m�XYZ = 85º
Calculate the area of each sector. Use 3.14 for �.
5. Find the area of the sector of a circle with a radius of 4 meters formed by a central
angle of 45º.
A � 45 ____ 360
�r2
A � 1 __ 8 �(4)2
A � ( 1 __ 8 ) (3.14)(16)
A � 6.28 square meters
6. Find the area of the sector of a circle with a radius of 12 meters formed by a central
angle of 30º.
A � 30 ____ 360
�r2
A � 1 ___ 12
�(12)2
A � ( 1 ___ 12
) (3.14)(144)
A � 37.68 square meters
7. Find the area of the sector of a circle with a radius of 22 inches formed by a central
angle of 60°.
A � 60 ____ 360
�r2
A � 1 __ 6 �(22)2
A � ( 1 __ 6 ) (3.14)(484)
A � 253.29 square inches
� �
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8. Find the area of the sector of a circle with a radius of 50 inches formed by a central
angle of 40°.
A � 40 ____ 360
�r2
A � 1 __ 9 �(50)2
A � ( 1 __ 9 ) (3.14)(2500)
A � 872.22 square inches
Calculate the length of each arc. Leave your answers in terms of �.
9. Find the length of an arc of a circle with a radius of 16 centimeters formed by a
central angle of 60º.
arc length � 60 ____ 360
2�r
arc length � 1 __ 6 2�(16)
arc length � 16 ___ 3 � centimeters
10. Find the length of an arc of a circle with a radius of 30 centimeters formed by a
central angle of 40º.
arc length � 40 ____ 360
2�r
arc length � 1 __ 9 2�(30)
arc length � 20 ___ 3 � centimeters
11. Find the length of an arc of a circle with a radius of 9 feet formed by a central angle of 50°.
arc length � 50 ____ 360
2�r
arc length � 5 ___ 36
2�(9)
arc length � 5 __ 2 � feet
12. Find the length of an arc of a circle with a radius of 62 feet formed by a central angle of 75°.
arc length � 75 ____ 360
2�r
arc length � 5 ___ 24
2�(62)
arc length � 155 ____ 6 � feet
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Answer each question using the given measurements. Use 3.14 for �. Round to the nearest hundredth, if necessary.
13. A 6-inch long pendulum swings through an angle of 90º every second. How far
does the tip of the pendulum move each second?
arc length � 90 ____ 360
2�r
arc length � ( 1 __ 4 ) 2�(6)
arc length � 3� � 9.42 inches
The tip of the pendulum moves about 9.42 inches each second.
14. The minute-hand on a clock is 4 inches long. How far does the tip of the
minute-hand move in 25 minutes?
In 25 minutes, the minute-hand on a clock moves 25 ___ 60
, or 5 ___ 12
of the way around the face, or 150º.
arc length � 150 ____ 360
2�r
arc length � ( 5 ___ 12
) 2�(4)
arc length � 10 ___ 3 � � 10.47 inches
The tip of the minute-hand moves about 10.47 inches in 25 minutes.
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15. A water sprinkler sprays water a distance of 30 feet. It rotates through a 120º angle.
What area of the lawn receives water?
30 ft
120°
The area receiving water is a sector of a circle.
A � 120 ____ 360
�r2
A � 1 __ 3 �(30)2
A � ( 1 __ 3 ) (3.14)(900)
A � 942 square feet
About 942 square feet of lawn receives water from the sprinkler.
16. A semicircular silk fan has a radius of 10 inches. Not including any overlap for
seams or edges, how much silk is used for the fan?
10 inches
The fan is a sector of a circle, with an angle of 180º.
A � 180 ____ 360
�r2
A � 1 __ 2 �(10)2
A � ( 1 __ 2 ) (3.14)(100)
A � 157 square inches
About 157 square inches of silk is used for the fan.
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Use the given information to determine the measure of each arc.
17.
P
T
S
R X
18.
A
D
CB
E
m�SXR � 40º m�ABD � 50º
mSR � 20º mAD � 68º
mPT � 100º mCE � 32º
m�SXR � ( 1 __ 2 ) (mPT � mSR) m�ABD � ( 1 __
2 ) (m AD � mCE )
40º � ( 1 __ 2 ) (mPT � 20º) 50º � ( 1 __
2 ) (68º � mCE )
80º � mPT � 20º 100º � 68º � mCE
100º � mPT 32º � mCE
19.
F
G
H
J
K
20.
V
W
X
Y
Z
m�GHF � 60º m�WXY � 70º
mGF � 32º mWY � 28º
mJK � 88º mVZ � 168º
m�GHF � 1 __ 2 (mGF � mJK ) m�WXY � 1 __
2 (mVZ � mWY )
60º � 1 __ 2 (32º � mJK ) 70º � 1 __
2 (mVZ � 28º )
120º � 32º � mJK 140º � mVZ � 28º
88º � mJK 168º � mVZ
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