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Chapter 11 ● Skills Practice 807
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Skills Practice Skills Practice for Lesson 11.1
Name _____________________________________________ Date ____________________
Riding a Ferris Wheel Introduction to Circles
VocabularyIdentify an instance of each term in the diagram.
1. circle
B
U
H
X T
I
A V
N
PM
2. center of the circle
3. chord of the circle
4. diameter
5. secant of the circle
6. tangent of the circle
7. point of tangency
8. central angle
9. inscribed angle
10. arc
11. major arc
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12. minor arc
Problem Set
Identify the indicated part of each circle. Explain your answer.
1. ___
OA 2. ___
GE
A
O B
C
E
FG
O
____
OA is a radius. It is a line segment that connects the center O to a point on the circle, A.
3. O 4. ___
NP
IH
J
K
O
N
M
P
O
5. ‹
___
› AB 6. D
A
O
C
B
E
D
F
O
Chapter 11 ● Skills Practice 809
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7. ___
JH 8. ‹
____
› MN
H
J
KI
O
L
MO
N
P
9. �SQR 10. �TOU
O
Q
RS
T
O
U
V
W
Use circle P to answer each question.
11. Which point(s) are located in the interior of
P
M
A
T
Y
Rthe circle?
Point P is in the interior of circle P.
12. Which point(s) are located on the circle?
13. Which point(s) are located in the exterior of
the circle?
14. Why isn’t ____
MR a diameter?
15. How are ____
MR and ‹
___ › AR alike?
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16. How are ____
MR and ‹
___ › AR different?
Identify each angle as an inscribed angle or a central angle.
17. �URE
O K
M
EU
Z
R
Angle URE is an inscribed angle.
18. �ZOM
19. �KOM
20. �ZKU
21. �MOU
22. �ROK
Classify each arc as a major arc, a minor arc, or a semicircle.
23. ⁀ AC 24. ⁀ DE
O
A
CB
E
F
O
D
⁀ AC is a minor arc.
25. ⁀ FHI 26. ⁀ JML
G
F
O
I
H
O
L
M
J
K
Chapter 11 ● Skills Practice 811
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27. ⁀ NPQ 28. ⁀ TRS
P
N
O
Q
S
R
U
T
O
Draw the part of a circle that is described.
29. Draw chord ___
AB . 30. Draw radius ___
OE .
O
A
B
O
31. Draw secant ‹
___
› GH . 32. Draw a tangent at point J.
O O
J
33. Label the point of tangency A. 34. Label center C.
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35. Draw inscribed angle �FDG. 36. Draw central angle �HOI.
O O
37. Draw major arc ⁀ KNM . 38. Draw minor arc ⁀ RQ .
O O
Chapter 11 ● Skills Practice 813
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Skills Practice Skills Practice for Lesson 11.2
Name _____________________________________________ Date ____________________
Holding the Wheel Central Angles, Inscribed Angles, and Intercepted Arcs
VocabularyDefine each term in your own words.
1. degree measure of a minor arc
2. adjacent arcs
3. Arc Addition Postulate
4. intercepted arc
5. Inscribed Angle Theorem
6. Parallel Lines-Congruent Arcs Theorem
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Problem SetDetermine the measure of each minor arc.
1. ⁀ AB 2. ⁀ CD
A
O 90� B
C
O 60� D
The measure of ⁀ AB is 90º.
3. ⁀ EF 4. ⁀ GH
O E45�
F
O H135�
G
5. ⁀ IJ 6. ⁀ KL
O I120�
J
K O85�
L
Chapter 11 ● Skills Practice 815
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Determine the measure of each central angle.
7. m�XYZ 8. m�BGT
30°
50°
Y Z
PX
40°
110°
GT
R
B
m�XYZ � 80°
9. m�LKJ 10. m�FMR
98°30°
K J
M
L
72°
31°
NM
R
F
11. m�KWS 12. m�VIQ
22°
48°
W
K N
S
85°
95°
I
V
Q
T
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Determine the measure of each inscribed angle.
13. m�XYZ 14. m�MTU
O
Z
Y
X
150°
82°
M
T Q
U
m�XYZ � 75°
15. m�KLS 16. m�DVA
K LO
S
112°
O
D A
V
86°
17. m�QBR 18. m�SGI
O
R
Q
B
155°S
G
OI
28°
Chapter 11 ● Skills Practice 817
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Determine the measure of each intercepted arc.
19. m ⁀ KM 20. m ⁀ IU
O
L
M
K
54°
L
I
O
U
36°
m ⁀ KM � 108°
21. m ⁀ QW 22. m ⁀ TV
Q
C
O
W
81°
A
T O
V
31°
23. m ⁀ ME 24. m ⁀ DS
M
R
O
E
52°SD O
P
90°
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Calculate the measure of each angle.
25. The measure of �AOB is 62º. What is the measure of �ACB?
O
C
A
B
m�ACB � 1 __ 2
(m�AOB) � 62º ____ 2
� 31º
26. The measure of �COD is 98º. What is the measure of �CED?
C
OED
27. The measure of �EOG is 128º. What is the measure of �EFG?
FO
E
G
Chapter 11 ● Skills Practice 819
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28. The measure of �GOH is 74º. What is the measure of �GIH?
G
O
I
H
29. The measure of �JOK is 168º. What is the measure of �JIK?
I
K
OJ
30. The measure of �KOL is 148º. What is the measure of �KML?
M
LK
O
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Use the given information to answer each question.
31. In circle C, m ⁀ XZ � 86º. What is m ⁀ WY ?
W
X
C
Y
Z
m ⁀ WY � 86°
32. In circle C, m�WCX � 102º. What is m ⁀ YZ ?
W Y
C
Z
X
33. In circle C, m ⁀ WZ � 65º and m ⁀ XZ � 38º. What is m�WCX?
C
W
Z
X
Chapter 11 ● Skills Practice 821
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34. In circle C, m�WCX � 105º. What is m�WYX?
W C
X
Y
35. In circle C, m�WCY � 83º. What is m�XCZ ?
WC
X
Z
Y
36. In circle C, m�WYX � 50º and m�XYZ � 30º. What is m ⁀ WXZ ?
W
C
Y
Z
X
Chapter 11 ● Skills Practice 823
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Skills Practice Skills Practice for Lesson 11.3
Name _____________________________________________ Date ____________________
Manhole Covers Measuring Angles Inside and Outside of Circles
VocabularyIdentify the similarities and differences between the terms.
1. Interior Angles of a Circle Theorem and Exterior Angles of a Circle Theorem
Problem SetList the two intercepted arcs for the given angle.
1. �AEB 2. �MRN
A
O
B
C
D
E
M
O
QP
N
R
⁀ AB , ⁀ CD
3. �VYU 4. �KNL
X
O
W
V
U
Y
J
O
M
L
K
N
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5. �EIF 6. �SUT
O G
HI
E
F
O
S
T
U
Q
R
Write an expression for the measure of the given angle.
7. m�RPM 8. m�ACD
O
N
M
P
Q
R O
E
B
C
D
A
m�RPM � 1 __ 2 ( m ⁀ RM � m ⁀ QN )
9. m�JNK 10. m�UWV
OK
JN
M
L
O
Y
X
WV
U
Chapter 11 ● Skills Practice 825
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11. m�SWT 12. m�HJI
O
T
V
W
U
S
O
I
G
J
H
F
List the intercepted arc(s) for the given angle.
13. �QMR 14. �RSU
O
R
Q
P
MN
O
P
S
U
T
R
⁀ NP , ⁀ QR
15. �FEH 16. �ZWA
O
F
H
EI
G OZ
W
A
YX
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17. �BDE 18. �JLM
O
C
EZ
D
B
O
L
MK
Z
J
Write an expression for the measure of the given angle.
19. m�DAC 20. m�UXY
O
C
BE
D
A
O
ZU
Y
X
W
m�DAC � 1 __ 2
( m ⁀ DEC � m ⁀ BC )
21. m�SRT 22. m�FJG
O
UR
S
V T
O
FG
IH
J
Chapter 11 ● Skills Practice 827
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23. m�ECG 24. m�LPN
O
DCE
H
G
F
OL
N p Q
M
Create a two-column proof to prove each statement.
25. Given: Chords ___
AE and ___
BD intersect at point C.
Prove: m�ACB � 1 __ 2 ( m ⁀ AB � m ⁀ DE )
O
A
D
E
BC
Statements Reasons
1. Chords ___
AE and ____
BD intersect at point C.
1. Given
2. Draw chord ___
AD 2. Construction
3. m�ACB � m�D � m�A 3. Exterior Angle Theorem
4. m�A � 1 __ 2 m ⁀ DE 4. Inscribed Angle Theorem
5. m�D � 1 __ 2 m ⁀ AB 5. Inscribed Angle Theorem
6. m�ACB � 1 __ 2 m ⁀ DE � 1 __
2 m ⁀ AB 6. Substitution
7. m�ACB � 1 __ 2 ( m ⁀ AB � m ⁀ DE ) 7. Distributive Property
828 Chapter 11 ● Skills Practice
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26. Given: Secant ‹
___
› QT and tangent
‹
___
› SR intersect at point S.
Prove: m�QSR � 1 __ 2 ( m ⁀ QR � m ⁀ RT )
O
Q
P
RT
S
Chapter 11 ● Skills Practice 829
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27. Given: Tangents ‹
___
› VY and
‹
___
XY intersect at point Y.
Prove: m�Y � 1 __ 2 ( m ⁀ VWX � m ⁀ VX )
O
Y
X
B
VA
W
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28. Given: Chords __
FI and ____
GH intersect at point J.
Prove: m�FJH � 1 __ 2 ( m ⁀ FH � m ⁀ GI )
O
GF
H
J
I
Chapter 11 ● Skills Practice 831
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29. Given: Secant ‹
___
› JL and tangent
‹
___
NL intersect at point L.
Prove: m�L � 1 __ 2
( m ⁀ JM � m ⁀ KM )
O
J
M
N
KL
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30. Given: Tangents ‹
___
› SX and
‹
___
› UX intersect at point X.
Prove: m�X � 1 __ 2 ( m ⁀ VTW � m ⁀ VW )
O
W
U
T
S
V
X
Chapter 11 ● Skills Practice 833
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11
Use the diagram shown to determine the measure of each angle or arc.
31. Determine m ⁀ FI . 32. Determine m�KLJ.
m�K � 20° m ⁀ KM � 120°
m ⁀ GJ � 80° m ⁀ JN � 100°
O
G
JK
F
I
O
KJ
NM
L
m ⁀ FI � 120°
33. Determine m�X. 34. Determine m�WYX.
m ⁀ VW � 50° m ⁀ WUY � 300°
m ⁀ TU � 85°
O
U
SR
T
V
X
W
O
W
U
ZYX
834 Chapter 11 ● Skills Practice
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35. Determine m ⁀ RS . 36. Determine m�D.
m ⁀ UV � 30° m ⁀ ZXC � 150°
m�RTS � 80° m ⁀ CB � 30°
O
R S
T
VU
O
Z
X
CB
A
D
Chapter 11 l Skills Practice 835
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Skills Practice Skills Practice for Lesson 11.4
Name _____________________________________________ Date ____________________
Color Theory Chords
VocabularyMatch each definition with its corresponding term.
1. Diameter-Chord Theorem a. If two chords of the same circle or
congruent circles are congruent, then their
corresponding arcs are congruent.
2. Equidistant Chord Theorem b. The segments formed on a chord when two
chords of a circle intersect.
3. Equidistant Chord c. If two chords of the same circle or congruent
Converse Theorem circles are congruent, then they are
equidistant from the center of the circle.
4. Congruent Chord-Congruent d. If two arcs of the same circle or congruent
Arc Theorem circles are congruent, then their
corresponding chords are congruent.
5. Congruent Chord-Congruent Arc e. If two chords of the same circle or congruent
Converse Theorem circles are equidistant from the center of the
circle, then the chords are congruent.
6. segments of a chord f. If two chords of a circle intersect, then the
product of the lengths of the segments of one
chord is equal to the product of the lengths of
the segments in the second chord.
7. Segment-Chord Theorem g. If a diameter of a circle is perpendicular to a
chord, then the diameter bisects the chord
and bisects the arc determined by the chord.
836 Chapter 11 ● Skills Practice
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Problem Set
Use the given information to answer each question. Explain your answer.
1. If diameter ___
BD bisects ___
AC , what is the angle of intersection?
D
B
CA
O
The angle of intersection is 90º because diameters that bisect chords are perpendicular bisectors.
2. If diameter ___
FH intersects ___
EG at a right angle, how does the length of __
EI compare to
the length of ___
IG ?
H
I
OGE
F
3. How does the measure of ⁀ KL and ⁀ LM compare?
O
N
K
L
M
J
Chapter 11 ● Skills Practice 837
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4. If ___
KP � ___
LN , how does the length of ____
QO compare to the length of ___
RO ?
O
Q RM
LK
P
J
N
5. If ___
YO � ___
ZO , what is the relationship between ___
TU and ___
XV ?
O
Y
S
T
Z
XW
V
U
6. If ____
GO � ____
HO and diameter ___
EJ is perpendicular to both, what is the relationship
between ___
GF and ___
HK ?
O
G
E
FD
I
J
HK
838 Chapter 11 ● Skills Practice
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Determine each measurement.
7. If ___
BD is a diameter, what is the length of ___
EC ?
5 cm
D
CAE
B
O
EC � EA � 5 cm
8. If the length of ___
AB is 13 millimeters, what is the length of ___
CD ?
D
B
A
CO
9. If the length of ___
AB is 24 centimeters, what is the length of ___
CD ?
A
O
D
C
B
10. If the length of ___
BF is 32 inches, what is the length of ___
CH ?
D
B
A
FG
O
CE H
Chapter 11 ● Skills Practice 839
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11. If the measure of �AOB � 155º, what is the measure of �DOC?
BA
O
D
C
12. If segment ___
AC is a diameter, what is the measure of �AED?
A
C
D
BE
O
Compare each measurement. Explain your answer.
13. If ___
DE � ___
FG , how does the measure of ⁀ DE and ⁀ FG compare?
O
D
F G
E
The measure of ⁀ DE is equal to the measure of ⁀ FG because congruent segments on the same circle create congruent chords.
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14. If ⁀ KM � ⁀ JL , how does the measure of ___
JL and ____
KM compare?
K
J
L
M
15. If ___
QR � ___
PS , how does the measure of ⁀ QPR and ⁀ PRS compare?
O
Q
P
R S
16. If ⁀ EDG � ⁀ DEH , how does the measure of ___
EG and ___
DH compare?
OF
E
D
G
H
Chapter 11 ● Skills Practice 841
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17. If �AOB � �DOC, what is the relationship between ___
AB and ___
DC ?
BA
OD
C
18. If �EOH � �GOF, what is the relationship between ⁀ EH and ⁀ FG ?
E
HO
F
G
Determine each measurement.
19. If ⁀ AB is congruent to ⁀ CD , what is the length of ___
CD ?
17 cm BA
C
D
O
CD � AB � 17 cm
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20. If ⁀ EF is congruent to ⁀ GH , what is the length of ___
EF ?
9 in.
F
E
OG
H
21. If ⁀ XU � ⁀ YV , what is the measure of ⁀ XU ?
O
Y
X
U
V
75°
22. If ___
AB � ___
CD , what is the measure of ⁀ AB ?
O
B
D
C
80°
A
23. If the length of ___
AB is 24 centimeters, what is the length of ___
CD ?
A
O
D
C
B
Chapter 11 ● Skills Practice 843
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24. If the length of ___
EF is 14 millimeters, what is the length of ___
HG ?
FE
H
O
G
Use the given circles to answer each question.
25. Name the segments of chord ___
KL .
O
N
LM
K J
____ KN and
___ NL
26. Name the segments of chord ___
MJ .
27. Name the segments of chord ___
RU .
OT
R
S
U
V
28. Name the segments of chord ___
SV .
29. Name the segments of chord ___
AC .
O E
B
A
CD
30. Name the segments of chord ___
DB .
844 Chapter 11 ● Skills Practice
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Use each diagram and the Segment Chord Theorem to write an equation involving the segments of the chords.
31.
OG
H J
D
F 32.
OQL
P
N
R
DG � GJ � FG � GH
33.
O
K
AB
VT
34.
OV
CS
GH
35.
O
Y
E
I
U
A
36.
O
J
X
L V
C
Chapter 11 ● Skills Practice 845
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Skills Practice Skills Practice for Lesson 11.5
Name _____________________________________________ Date ____________________
Solar Eclipses Tangents and Secants
VocabularyWrite the term from the box that best completes each statement.
external secant segment Secant-Segment Theorem tangent segment
secant segment Secant-Tangent Theorem Tangent-Segment Theorem
1. A(n) is the segment that is formed from an exterior
point of a circle to the point of tangency.
2. The states that if two tangent segments are drawn
from the same point on the exterior of the circle, then the tangent segments
are congruent.
3. When two secants intersect in the exterior of a circle, the segment that begins at
the point of intersection, continues through the circle, and ends on the other side
of the circle is called a(n) .
4. When two secants intersect in the exterior of a circle, the segment that begins at
the point of intersection and ends where the secant enters the circle is called a(n)
.
5. The states that if two secants intersect in the
exterior of a circle, then the product of the lengths of the secant segment and
its external secant segment is equal to the product of the lengths of the second
secant segment and its external secant segment.
6. The states that if a tangent and a secant intersect
in the exterior of a circle, then the product of the lengths of the secant segment
and its external secant segment is equal to the square of the length of the
tangent segment.
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Problem Set
Calculate the measure of each angle. Explain your reasoning.
1. If ___
OA is a radius, what is the measure of �OAB?
O
A
B
A tangent line and the radius that ends at the point of tangency are perpendicular, so m�OAB � 90º.
2. If ____
OD is a radius, what is the measure of �ODC?
C
D
O
3. If ___
YO is a radius, what is the measure of � XYO?
O
Y
X
Chapter 11 ● Skills Practice 847
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4. If ___
RS is a tangent segment and ___
OS is a radius, what is the measure of �ROS?
35°R O
S
5. If ___
UT is a tangent segment and ____
OU is a radius, what is the measure of �TOU?
23°O
U
T
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6. If ____
V W is a tangent segment and ___
OV is a radius, what is the measure of �VWO?
72°O
V
W
Write a statement to show the congruent segments.
7. ‹
___
› AC and
‹
___
› BC are tangent to circle O. 8.
‹
___
› XZ and
‹
___
› ZW are tangent to circle O.
O
A
C
B
O
Z
XW
___
AC � ____
CB
9. ‹
___
› RS and
‹
___
› RT are tangent to circle O. 10.
‹
___
› MP and
‹
___
› NP are tangent to circle O.
O
S
R
T
O
M
P
N
Chapter 11 ● Skills Practice 849
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11
11. ‹
___
› DE and
‹
___
› FE are tangent to circle O. 12.
‹
___
› GH and
‹
__
› GI are tangent to circle O.
O
D
E
F
O
H
G
I
Calculate the measure of each angle. Explain your reasoning.
13. If ___
EF and ___
GF are tangent segments, what is the measure of �EGF?
O
E
F
G
64°
Because ___
EF and ___
GF are tangent segments drawn from point F, they must be congruent. This fact means that � EFG is an isosceles triangle.
m�EGF � (180º � 64º) � 2 � 116º � 2 � 58º
The measure of �EGF is 58º.
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14. If ___
HI and __
JI are tangent segments, what is the measure of �HJI?
H
I48°
O
J
15. If ____
KM and ___
LM are tangent segments, what is the measure of �KML?
K
OM
63°
L
Chapter 11 ● Skills Practice 851
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16. If ___
NP and ___
QP are tangent segments, what is the measure of �NPQ?
P
N
O
Q
71°
17. If ___
AF and ___
VF are tangent segments, what is the measure of � AVF?
O
A
F
V
22°
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18. If ___
RT and ___
MT are tangent segments, what is the measure of �RTM?
O
M
T
R
33°
Name two secant segments and two external secant segments for circle O.
19.
O
RT
S
QP
20.
OB
A
C
DE
Secant segments: ___
PT and ___
QT
External secant segments: ___
RT and ___
ST
21.
O
U V
X
Y
W 22.
O
L
R
M
P
N
Chapter 11 ● Skills Practice 853
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Name _____________________________________________ Date ____________________
11
23.
OH
J
IG
F 24.
O
J
K
L
M
N
Use each diagram and the Secant Segment Theorem to write an equation involving the secant segments.
25.
O
S
R
T
U
V
26.
ON
M
QR
S
RV � TV � SV � UV
27.
O
DBA
C
E
28.
O
J
H
LK
I
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29.
O
F
E
B
CA
30.
O
Y Z
X
W
V
Name a tangent segment, a secant segment, and an external secant segment for circle O.
31.
O
R
S T
U
32.
O
JH
K
I
Tangent segment: ___
TU
Secant segment: ___
RT
External secant segment: ___
ST
33.
O
GF
D
E 34.
O
S
R
Q
P
Chapter 11 ● Skills Practice 855
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35.
OE
GF
H
36.
O
y
X
Z
W
Use each diagram and the Secant Tangent Theorem to write an equation involving the secant and tangent segments.
37.
O W
M
E
Q
38.
O
S
RV
B
(EM )2 � QM � WM
39.
X
ZG
I
40.
O
TBA
X
856 Chapter 11 ● Skills Practice
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41.
O
EV
W
U 42.
O
M
l
C
Q
Chapter 11 ● Skills Practice 857
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Skills Practice Skills Practice for Lesson 11.6
Name _____________________________________________ Date ____________________
Replacement for a Carpenter’s Square Inscribed and Circumscribed Triangles and Quadrilaterals
VocabularyAnswer each question.
1. How are inscribed polygons and circumscribed polygons different?
2. Describe how you can use the Inscribed Right Triangle-Diameter Theorem to show
an inscribed triangle is a right triangle.
3. What does the Converse of the Inscribed Right Triangle-Diameter Theorem help to
show in a circle?
4. What information about a quadrilateral inscribed in a circle does the Inscribed
Quadrilateral-Opposite Angles Theorem give?
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Problem SetDraw a triangle inscribed in the circle through the three points. Then determine if the triangle is a right triangle.
1. A
BC
O
2.
A
B
C
O
No. The triangle is not a right triangle. None of the sides of the triangle is a diameter of the circle.
3. A
B
C
O
4. A
B
C
O
5. A
C
B
O
6.
C
A
B
O
Chapter 11 ● Skills Practice 859
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11
7.
C
A
B
O
8.
C
A
B
O
Draw a triangle inscribed in the circle through the given points. Then determine the measure of the indicated angle.
9. In � ABC, m�A � 55°. Determine m�B.
A
C
B
m�B � 180° � 90° � 55° � 35°
10. In � ABC, m�B � 38°. Determine m�A.
B
C
A
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11. In � ABC, m�C � 62°. Determine m� A.
A
B
C
12. In � ABC, m�B � 26°. Determine m�C.
C
A
B
13. In � ABC, m�C � 49°. Determine m� A.
A
C
B
14. In � ABC, m�B � 51°. Determine m�A.
A
C
B
Chapter 11 ● Skills Practice 861
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11
Draw a quadrilateral inscribed in the circle through the given four points. Then determine the measure of the indicated angle.
15. In quadrilateral ABCD, m�B � 81°. Determine m�D.
A
B
C
D
m�D � 180° � 81° � 99°
16. In quadrilateral ABCD, m�C � 75°. Determine m�A.
A
B
C
D
17. In quadrilateral ABCD, m�B � 112°. Determine m�D.
A B
CD
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18. In quadrilateral ABCD, m�D � 93°. Determine m�B.
A B
CD
19. In quadrilateral ABCD, m�A � 72°. Determine m�C.
A
B
C
D
20. In quadrilateral ABCD, m�B � 101°. Determine m�D.
A
B
C
D
Chapter 11 ● Skills Practice 863
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11
Construct a circle inscribed in each polygon.
21.
22.
Chapter 11 ● Skills Practice 865
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26.
Create a two-column proof to prove each statement.
27. Given: Inscribed � ABC in circle O, m ⁀ AC � 40°, and m ⁀ BC � 140°
Prove: ___
AB is a diameter of circle O.
A
B
C
40°
140°
O
Statements Reasons
1. m ⁀ AC � 40°, m ⁀ BC � 140° 1. Given
2. m ⁀ AC � m ⁀ BC � m ⁀ AB � 360° 2. Arc Addition Postulate
3. 40° � 140° � m ⁀ AB � 360° 3. Substitution
4. m ⁀ AB � 180° 4. Subtraction Property of Equality
5. m�C � 1 __ 2 m ⁀ AB 5. Definition of inscribed angle
6. m�C � 90° 6. Substitution
7. � ABC is a right triangle with right angle C. 7. Definition of right triangle
8. ___
AB is the diameter of circle O. 8. Converse of Inscribed Right Triangle-Diameter Theorem
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28. Given: Inscribed �RST in circle O with diameter ___
RS , and m ⁀ RT � 60°
Prove: m ⁀ ST � 120°
T
S
60°
R
Chapter 11 ● Skills Practice 867
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11
29. Given: Inscribed quadrilateral ABCD in circle O, m ⁀ AB � 50°, and m ⁀ BC � 90°
Prove: m�B � 110°
90°
50°
C
B
A
D
O
868 Chapter 11 ● Skills Practice
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30. Given: Inscribed quadrilateral MNPQ in circle O, m ⁀ MQ � 75°, and m�NMQ � 120°
Prove: m ⁀ MN � 45°
120°75°
QP
NM
O
Chapter 11 ● Skills Practice 869
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11
31. Given: m ⁀ AB � 50°, m ⁀ BC � 90°, and m ⁀ CD � 90°
Prove: m�BCD � 90°
B
D
A
C
870 Chapter 11 ● Skills Practice
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32. Given: �BAD and �ADC are supplementary angles
Prove: m�BAD � m�ABC
B
C
A
D
Chapter 11 ● Skills Practice 871
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Skills Practice Skills Practice for Lesson 11.7
Name _____________________________________________ Date ____________________
Gears Arc Length
VocabularyDefine the key term in your own words.
1. arc length
Problem SetCalculate the ratio of the length of each arc to the circle’s circumference.
1. The measure of ⁀ AB is 40º. 2. The measure of ⁀ CD is 90º.
40º _____ 360º
� 1 __ 9
The arc is 1 __ 9 of the circle’s
circumference.
3. The measure of ⁀ EF is 120º. 4. The measure of ⁀ GH is 150º.
5. The measure of ⁀ IJ is 105º. 6. The measure of ⁀ KL is 75º.
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Write an expression that you can use to calculate the length of ⁀ RS . You do not need to simplify the expression.
7. 80°
15 in. S
R 8.
110°
13 cm
R
S
80 ____ 360
� 2�(15)
9.
90°
17 cmR
S
10.
160°
28 cm
R
S
11.
180°
63 ft
R
S
12.
150°
33 yd
R
S
Calculate each arc length. Write your answer in terms of �.
13. If the measure of ⁀ AB is 45º and the radius is 12 meters, what is the arc length of ⁀ AB ?
C � 2�(12) � 24�
Fraction of C: 45º _____ 360º
� 1 __ 8
Arc length of ⁀ AB : 1 __ 8 (24�) � 3�
The arc length of ⁀ AB is 3� meters.
Chapter 11 ● Skills Practice 873
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11
14. If the measure of ⁀ CD is 120º and the radius is 15 centimeters, what is the arc
length of ⁀ CD ?
15. If the measure of ⁀ EF is 60º and the radius is 8 inches, what is the arc length of ⁀ EF ?
16. If the measure of ⁀ GH is 30º and the radius is 6 meters, what is the arc length of ⁀ GH ?
17. If the measure of ⁀ IJ is 80º and the diameter is 10 centimeters, what is the arc length of ⁀ IJ ?
18. If the measure of ⁀ KL is 15º and the diameter is 18 feet, what is the arc length of ⁀ KL ?
874 Chapter 11 ● Skills Practice
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19. If the measure of ⁀ MN is 75º and the diameter is 20 millimeters, what is the arc length
of ⁀ MN ?
20. If the measure of ⁀ OP is 165º and the diameter is 21 centimeters, what is the arc
length of ⁀ OP ?
Calculate each arc length. Write your answer in terms of �.
21. If the measure of ⁀ AB is 135º, what is the arc length of ⁀ AB ?
A
16 cm
B
C � 2�(16) � 32�
Fraction of C: 135º _____ 360º
� 3 __ 8
Arc length of ⁀ AB : 3 __ 8 (32�) � 96 ___
8 � � 12�
The arc length of ⁀ AB is 12� cm.
Chapter 11 ● Skills Practice 875
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11
22. If the measure of ⁀ CD is 45º, what is the arc length of ⁀ CD ?
C
D
18 cm
23. If the measure of ⁀ EF is 90º, what is the arc length of ⁀ EF ?
E
F30 in.
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24. If the measure of ⁀ GH is 120º, what is the arc length of ⁀ GH ?
G
27 in.
H
25. If the length of the radius is 4 centimeters, what is the arc length of ⁀ IJ ?
I
160°
J
Chapter 11 ● Skills Practice 877
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11
26. If the length of the radius is 7 centimeters, what is the arc length of ⁀ KL ?
20°
K L
27. If the length of the radius is 11 centimeters, what is the arc length of ⁀ MN ?
100°
M
N
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28. If the length of the radius is 17 centimeters, what is the arc length of ⁀ OP ?
75° P
O
Use the given information to answer each question. Where necessary, use 3.14 to approximate �.
29. Determine the perimeter of the shaded region.
30 mm 60˚
Arc length: 60 ____ 360
� 2(3.14)(30) � 31.4 mm
Perimeter of shaded region: 31.4 � 30 � 30 � 91.4 mm
Chapter 11 ● Skills Practice 879
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11
30. Determine the perimeter of the figure below.
180°30 cm
15 cm
31. A semicircular cut was taken from the rectangle shown. Determine the perimeter of
the shaded region.
60 ft
80 ft
32. A circle has a circumference of 81.2 inches. What is the radius of the circle?
880 Chapter 11 ● Skills Practice
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33. Bella used a tape measure and found the circumference of a flagpole to be
6.2 inches. What is the radius of the flagpole?
34. Carla used a string and a tape measure and found the circumference of a circular
cup to be 12.56 inches. What is the radius of the cup?
Chapter 11 ● Skills Practice 881
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Skills Practice Skills Practice for Lesson 11.8
Name _____________________________________________ Date ____________________
Playing Darts Sectors and Segments of a Circles
VocabularyDraw an example of each term.
1. concentric circles 2. sector of a circle
3. segment of a circle
Problem SetCalculate the area of each circle. Write your answer in terms of �.
1. What is the area of a circle whose radius is 5 centimeters?
A � �(52) � 25�
The area of the circle is 25� cm2.
2. What is the area of a circle whose radius is 8 millimeters?
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3. What is the area of a circle whose radius is 12 feet?
4. What is the area of a circle whose radius is 18 centimeters?
5. What is the area of a circle whose diameter is 22 inches?
6. What is the area of a circle whose diameter is 28 meters?
7. What is the area of a circle whose diameter is 15 inches?
8. What is the area of a circle whose diameter is 31 yards?
Chapter 11 ● Skills Practice 883
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11
Calculate the area of each sector. Write your answer in terms of �.
9. If the radius of the circle is 9 centimeters, what is the area of sector AOB?
120°
A
B
O
Total area of the circle � �(92) � 81� cm2
Sector AOB’s fraction of the circle � 120º _____ 360º
� 1 __ 3
Area of sector AOB � 1 __ 3 (81�) � 27�
The area of sector AOB is 27� cm2.
10. If the radius of the circle is 16 meters, what is the area of sector COD?
45°
C
O
D
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11. If the radius of the circle is 15 feet, what is the area of sector EOF?
30°
E
F
O
12. If the radius of the circle is 10 inches, what is the area of sector GOH?
20°
G H
O
Chapter 11 ● Skills Practice 885
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11
13. If the radius of the circle is 32 centimeters, what is the area of sector IOJ?
I
O
18°
J
14. If the radius of the circle is 20 millimeters, what is the area of sector KOL?
O L
K
80°
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15. If the radius of the circle is 24 centimeters, what is the area of sector MON?
O
M
135°
N
16. If the radius of the circle is 21 meters, what is the area of sector POQ?
P
Q
150° O
Chapter 11 ● Skills Practice 887
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Name _____________________________________________ Date ____________________
11
Calculate the area of each segment. Round your answer to the nearest tenth, if necessary. Use 3.14 to estimate �.
17. If the radius of the circle is 6 centimeters, what is the area of the shaded segment?
A
BO90°
Total area of the circle � �(62) � 36� cm2
Sector AOB’s fraction of the circle � 90º _____ 360º
� 1 __ 4
Area of sector AOB � 1 __ 4 (36�) � 9� cm2
Area of � AOB � 1 __ 2 (6 � 6) � 18 cm2
Area of the segment: 9� � 18 � 28.3 � 18 � 10.3
The area of the shaded segment is approximately 10.3 cm2.
18. If the radius of the circle is 14 inches, what is the area of the shaded segment?
OD
C
90°
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19. If the radius of the circle is 17 feet, what is the area of the shaded segment?
OE
F
90°
20. If the radius of the circle is 22 centimeters, what is the area of the
shaded segment?
OG
H
90°
Chapter 11 ● Skills Practice 889
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11
21. If the radius of the circle is 25 meters, what is the area of the shaded segment?
O
I J
90°
22. If the radius of the circle is 30 centimeters, what is the area of the shaded
segment?
O
LK
90°
890 Chapter 11 ● Skills Practice
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In circle O below, m ⁀ AB � 90°. Use the given information to determine the length of the radius of circle O.
A
Bo
23. If the area of the segment is 16� � 32 square feet, what is the length of the radius
of circle O?
16� � 90º _____ 360º
(�r2)
16� � 1 __ 4 (�r2)
64 � r2
8 � r
The length of the radius is 8 feet.
24. If the area of the segment is 25� � 50 square inches, what is the length of the
radius of circle O?
25. If the area of the segment is � � 2 square meters, what is the length of the radius
of circle O?
Chapter 11 ● Skills Practice 891
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11
26. If the area of the segment is 56.25� � 112.5 square yards, what is the length of the
radius of circle O?
27. If the area of the segment is 121� � 242 square feet, what is the length of the
radius of circle O?
28. If the area of the segment is 90.25� � 180.5 square millimeters, what is the length
of the radius of circle O?
Chapter 11 ● Skills Practice 893
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Skills Practice Skills Practice for Lesson 11.9
Name _____________________________________________ Date ____________________
The Coordinate Plane Circles and Polygons on the Coordinate Plane
Problem SetUse the given information to show that each statement is true. Justify your answers by using theorems and by using algebra.
1. The center of circle O is at the origin. The
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6
BA O
D
C
x
y
coordinates of the given points are A(�4, 0),
B(4, 0), and C(0, 4). Show that Δ ABC is a right
triangle.
� ABC is an inscribed triangle in circle O with the hypotenuse as the diameter of the circle, therefore the triangle is a right triangle by the Diameter-Right Triangle Converse Theorem.
AB � √___________________
(�4 � 4 ) 2 � (0 � 0 ) 2
� √_____
(�8 ) 2 � √___
64 � 8
AC � √___________________
(�4 � 0 ) 2 � (0 � 4 ) 2
� √_____________
(�4)2 � (�4)2
� √___
32 � 4 √__
2
BC � √_________________
(4 � 0 ) 2 � (0 � 4 ) 2
� √__________
42 � (�4)2
� √___
32 � 4 √__
2
(4 √__
2 ) 2 � (4 √__
2 ) 2 � 82
32 � 32 � 64
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2. The center of circle O is at the origin. Lines ‹
___
› AZ and
x10
10
20
30
–10
–10 0
–20
20 30–20–30
y
–30
T
Z
A ‹
___
› AT are tangent to circle O. The coordinates of the
given points are A(�10, 30), T(8, 6), and Z(�10, 0).
Show that the lengths of ___
AT and ___
AZ are equal.
3. The center of circle C is at the origin. Line ‹
___
› AB is
42
2
4
C (0, 0)
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
B (–1, 7)
A (3, 4)
x
y
tangent to circle C at (3, 4). Show that the tangent
line is perpendicular to ___
CA .
Chapter 11 ● Skills Practice 895
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4. The center of circle O is at the origin. The
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
10
–10
E
F
B
D
O
A
x
y
coordinates of the given points are A(3, 4),
B(�4, �3), D(0, �5), E(�3, 4), and F(�1.5, �0.5).
Show that EF � FD � AF � FB.
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5. The center of circle O is at the origin. The
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
10
–10
B
A D
O
C
x
y
coordinates of the given points are A(�5, 5),
B(�4, 3), C(0, �5), and D(0, 5). Show that
AD2 � AB � AC.
Chapter 11 ● Skills Practice 897
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6. The center of circle O is at the origin. Chord ___
BD is
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
10
–10
B
A
D
OC x
y
perpendicular to radius ___
OA at point C. The
coordinates of the given points are A(�5, 0),
B(�4, 3), C(�4, 0), and D(�4, �3). Show that
___
OA bisects ___
BD .
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Classify the polygon formed by connecting the midpoints of the sides of each quadrilateral. Show all your work.
7. The rectangle shown has vertices A(0, 0), B(x, 0),
D (0, y)
A (0, 0) B (x, 0)
C (x, y)
SR
U
T
C(x, y), and D(0, y).
Midpoint ___
AB : U ( x __ 2 , 0 )
Midpoint ____
BC : S ( x, y __
2 )
Midpoint ____
CD : T ( x __ 2
, y ) Midpoint
___ DA : R ( 0,
y __
2 )
Slope ___
RT � y �
y __
2 ______
x __ 2 � 0
� y __
2 __
x __ 2 �
y __ x Slope
___ TS �
y __
2 � y ______
x � x __ 2 �
� y __
2 ____
x __ 2 � �
y __ x
Slope ___
US � 0 �
y __
2 ______
x __ 2 � x
� �
y __
2 ____
� x __ 2 �
y __ x Slope
____ RU �
y __
2 � 0 ______
0 � x __ 2 �
y __
2 ____
� x __ 2 � �
y __ x
Sides ___
RT and ___
US are parallel since they have the same slope of y __ x .
Sides ___
TS and ____
RU are parallel since they have the same slope of � y __ x .
There are no perpendicular sides. The slopes are not negative reciprocals.
RT � √__________________
( 0 � x __ 2 ) 2 � ( y __
2 � y ) 2
TS � √__________________
( x __ 2 � x ) 2 � ( y �
y __
2 ) 2
� √_____________
( � x __ 2 ) 2 � ( �
y __
2 ) 2
� √____________
( � x __ 2 ) 2 � ( y __
2 ) 2
� √_______
x2 __
4 �
y2
__ 4 � √
_______
x2 __
4 �
y2
__ 4
US � √__________________
( x � x __ 2 ) 2 � ( y __
2 � 0 ) 2
RU � √__________________
( 0 � x __ 2 ) 2 � ( y __
2 � 0 ) 2
� √__________
( x __ 2 ) 2 � ( y __
2 ) 2
� √____________
( � x __ 2 ) 2 � ( y __
2 ) 2
� √_______
x2 __
4 �
y2
__ 4 � √
_______
x2 __
4 �
y2
__ 4
All four sides of the new quadrilateral are congruent.
Opposite sides are parallel and all sides are congruent, so the quadrilateral formed by connecting the midpoints of the rectangle is a rhombus.
Chapter 11 ● Skills Practice 899
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8. The isosceles trapezoid shown has vertices
A (0, 0)
D (2, 3) C (4, 3)
B (6, 0)
A(0, 0), B(6, 0), C(4, 3), and D(2, 3).
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9. The parallelogram shown has vertices A(4, 5),
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
10
–10
BA
DC
x
y
B(7, 5), C(3, 0), and D(0, 0).
Chapter 11 ● Skills Practice 901
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10. The rhombus shown has vertices A(4, 10), B(8, 5),
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
10
–10
B
A
D
Cx
y
C(4, 0), and D(0, 5).
902 Chapter 11 ● Skills Practice
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11. The square shown has vertices A(0, 0), B(4, 0),
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
10
–10
BA
DC
x
y
C(4, 4), and D(0, 4).
Chapter 11 ● Skills Practice 903
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12. The kite shown has vertices A(0, 2), B(2, 0),
42
2
4
6
8
–8–2
–2 0 86–4
–4
–6
–8
–6 10–10
10
–10
B
A
D
C
x
y
C(0, � 3), and D(�2, 0).
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