GA2K 10105 5.1

Chapter 5 l Skills Practice 399

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

400 Chapter 5 l Skills Practice

5

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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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

�

�

�

�

�

�


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


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.

� � � �

� �

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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°

� �

� �

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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°

� �

� �

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


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

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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°

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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°


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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°

�

�

� �

�

�

�

� �

�

�

� � �

�

�

��


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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°

� � � �

� � � �

�

�

�

�

�

�

�

�


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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°

� �

� �

�

�

�

�

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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°

�

�

�

�

� �

�

�

�

�

�

�

�

�

�


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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°

� �

� �

� � ��


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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°

� � � �

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


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 �

�

�

�

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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

� ��


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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.

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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.


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


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� �

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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

� �

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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

� �

��

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


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

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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

� �

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

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

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

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

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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 )

� √__


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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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.


� 1 __ 2 (2)( √

__ 3 )

� √__


Area of segment � 2.09 � 1.73 � 0.36 square centimeters


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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)


�PQR is an equilateral triangle, so the base is 6 centimeters and the height is 3 √

__ 3 centimeters.


� 1 __ 2 (6)(3 √

__ 3 )


Area of segment � 18.84 � 15.59 � 3.25 square centimeters


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5


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

� �

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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

� �


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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

© 2

009 C

arn

eg

ie L

earn

ing

, In

c.


5

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.


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

Name _____________________________________________ Date ____________________

5

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.


© 2

009 C

arn

eg

ie L

earn

ing

, In

c.

5

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

��

� �

��

��

� �

� �

� ��

� � � �

� �

� �

� �

GA2K 10105 5.1

Documents

Transcript of GA2K 10105 5.1