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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

808 Chapter 11 ● Skills Practice

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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


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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


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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


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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


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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°


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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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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.


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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


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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


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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


AB is 24 centimeters, what is the length of ___

CD ?

A

O

D

C

B


BF is 32 inches, what is the length of ___

CH ?

D

B

A

FG

O

CE H


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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


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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


AB is 24 centimeters, what is the length of ___

CD ?

A

O

D

C

B


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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


MJ .


RU .

OT

R

S

U

V


SV .


AC .

O E

B

A

CD


DB .


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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


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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


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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


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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


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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


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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


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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


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41.

O

EV

W

U 42.

O

M

l

C

Q


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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


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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


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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


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


A

B

C

D


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Construct a circle inscribed in each polygon.

21.

22.


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23.

24.

25.


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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


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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


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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


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31. Given: m ⁀ AB � 50°, m ⁀ BC � 90°, and m ⁀ CD � 90°

Prove: m�BCD � 90°

B

D

A

C


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32. Given: �BAD and �ADC are supplementary angles

Prove: m�BAD � m�ABC

B

C

A

D


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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.


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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 ?


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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.


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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


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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


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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?


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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?


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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?


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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


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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


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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°


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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°


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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?


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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?


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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 .


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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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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


B(�4, 3), C(0, �5), and D(0, 5). Show that

AD2 � AB � AC.


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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


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.


© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

11

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).


© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

11

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).


© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

11

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).


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010 C

arn

eg

ie L

earn

ing

, In

c.

11

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).


© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

11

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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