chapter 11 · chapter 11 worked-out solution key. chapter 11continued 188 geometry, concepts and...

CHAPTER 11

Geometry, Concepts and Skills 185Chapter 11 Worked-Out Solution Key

Copyright © McDougal Littell Inc. All rights reserved.

Chapter Opener

Chapter Readiness Quiz (p. 588)

1. D; 162 � 302 � 3422. F

256 � 900 � 1156

1156 � 1156

3. D; 5x � 8 � 2x � 7

3x � 15

x � 5

Lesson 11.1

11.1 Checkpoint (p. 590)

1. ED**** is a chord (as are JK**** and FH**** );

JK&̂( is a secant; AC&̂( is a tangent;

FH**** is a diameter; FG**** is a radius (as is GH**** );

G is the center; B is a point of tangency.

2. (0, 4)

11.1 Guided Practice (p. 591)

1. Sample answer:

2. B 3. E 4. D 5. A 6. C 7. F

8. (3, 3) 9. (3, 0) and (3, 6)

10. (0, 3), (3, 0), or (6, 3) 11. (3, 0) and (0, 3)

12. (3, 3) and one of the following: (3, 0), (6, 3), (3, 6)

11.1 Practice and Applications (pp. 591–593)

13. r � �12

�d � �12

�(15) � 7.5 cm

14. r � �12

�d � �12

�(6.5) � 3.25 in.

15. r � �12

�d � �12

�(3) � 1.5 ft 16. r � �12

�d � �12

�(8) � 4 m

17. d � 2r � 2(26) � 52 in. 18. d � 2r � 2(62) � 124 ft

19. d � 2r � 2(8.7) � 17.4 m 20. d � 2r � 2(4.4) � 8.8 cm

21. chord 22. tangent 23. diameter 24. radius

25. point of tangency 26. center of the circle

tangent

diameter

radius

chord

27. chord 28. radius 29. diameter 30. secant

31. tangent 32. secant

33. EG**** is a chord (as is EF**** );

EG&̂( is a secant;

EF**** is a diameter;

CE**** is a radius (as are CF**** and CG**& );

D is a point of tangency.

34. MN**& is a chord (as is JL**** );

MN^&( is a secant;

JL**** is a diameter;

KR**** is a radius (as are KJ**** and KL**** );

U is a point of tangency.

35. LM**** is a chord (as is PN**** );

LM&̂( is a secant;

PN**** is a diameter;

QR**** is a radius (as are QP**** and QN**** );

K is a point of tangency.

36. FA&̂( and EB&̂( are secants.

37. Any two of GD**&, HC**** , FA**** , and EB**** are chords.

38. Yes; the diameter is the longest chord and must passthrough the center of the circle. Since HC**** does not passthrough the center, it is shorter than the diameter.

39. Yes, a line can be drawn through points K, G, and J thatwill be tangent to the circle.

40. The center of �A is (2, 2); the center of �B is (6, 2).

41. The length of the radius of �A is 2 units; the length ofthe radius of �B is 2 units.

42. (4, 2)

43. �A: (3, 2); �B: (3, 3); intersection: (3, 0); tangent line: x-axis

44. �A: (3, 3); �B: (2, 3); intersection: (0, 3); tangent line: y-axis

45. r � 2; d � 4

46. AE2 � EB2 � AB247.

22 � 22 � AB2

4 � 4 � AB2

8 � AB2

2�2� � AB

2.8 � AB

11.1 Standardized Test Practice (p. 593)

48. D 49. G

y

x

1

1

B

CA

D

E(4, 3)

Chapter 11 continued

11.1 Mixed Review (p. 593)

50. HL Congruence Theorem; BC**** � BC**** and BA**** � BD**** inright triangles BAC and BDC, so T BAC �T BDC bythe HL Congruence Theorem.

51. SSS Congruence Postulate; three sides of T JKL arecongruent to three sides of T PQR, so T JKL �T PQRby the SSS Congruence Postulate.

52. ASA Congruence Postulate; aP � aT, PR**** � TR**** , andaPRQ � aTRS, so T PQR �T TSR by the ASACongruence Postulate.

53. slope of AB**** � �31

�

�

00

� � 3

slope of BC**** � �35

�

�

31

� � 0

slope of CD**** � �35

�

�

04

� � 3

slope of DA****� �04

�

�

00

� � 0

The opposite sides of ABCD have equal slopes, so theyare parallel. So, ABCD is a parallelogram.

54. slope of QR**** � �52

�

�

50

� � 0

slope of RS**** � �52

�

�

14

� � �2

slope of PS**** � �14

�

�

12

� � 0

slope of QP**** � �50

�

�

12

� � �2

The opposite sides of PQRS have equal slopes, so theyare parallel. So, PQRS is a parallelogram.

11.1 Algebra Skills (p. 593)

55. �3�2� � �1�6� �� 2� � 4�2� � 5.7

56. �8�1� � �9� �� 9� � 9

57. �4�0� � �4� �� 1�0� � 2�1�0� � 6.3

58. �1�0�4� � �4� �� 2�6� � 2�2�6� � 10.2

59. �9�8� � �4�9� �� 2� � 7�2� � 9.9

60. �1�9�2� � �6�4� �� 3� � 8�3� � 13.9

61. �2�5�0� � �2�5� �� 1�0� � 5�1�0� � 15.8

62. �2�4�2� � �1�2�1� �� 2� � 11�2� � 15.6

63. 2x � 5 � 19 64. 7x � 7 � 14

2x � 14 7x � 21

x � 7 x � 3

65. 5x � 9 � 4 66. 3x � 10 � 20

5x � �5 3x � 30

x � �1 x � 10

y

1

1

x

(0, 5)R(2, 5)

P(2, 1) S(4, 1)

y

4

4

x

B(1, 3) C(5, 3)

A(0, 0)D(4, 0)

67. 12 � 8x � 84 68. 4x � 3 � 23

�8x � 72 4x � 20

x � �9 x � 5

Lesson 11.2

11.2 Activity (p. 594)

1. You get the same results for each circle: MP � NP andmaCMP � maCNP � 90�.

2. The lengths are equal.

3. The angle is a right angle.

11.2 Checkpoint (p. 597)

1. CD � CB

x � 15

2. CB � CD

3x � 5 � 14

3x � 9

x � 3


1. AB&̂( is tangent to �C, and point B is the point of tangency.

2. 90�; If a line is tangent to a circle, then it is perpendicularto the radius drawn at the point of tangency.

3. No; 52 � 52 � 72, so T ABD is not a right triangle andAB**** is not perpendicular to BD**** . Therefore, BD**** is not tangent to �C.

4. AB � DB 5. AB � DB

x � 4 2 � x

6. AB � DB

2x � 10

x � 5

11.2 Practice and Applications (p. 598–600)

7. BA2 � AC2 � BC28. AB2 � AC2 � BC2

42 � r2 � 52 152 � r2 � 172

16 � r2 � 25 225 � r2 � 289

r2 � 9 r2 � 64

r � 3 r � 8

9. AB2 � AC2 � BC210. AB � AD

202 � r2 � 252 x � 7

400 � r2 � 625

r2 � 225

r � 15

186 Geometry, Concepts and SkillsChapter 11 Worked-Out Solution Key




11. AB � AD

x � 4 � 19

x � 15

12. AB � AD

2x � 7 � 5x � 8

�3x � �15

x � 5

13. (x � 2)2 � (x � 2)(x � 2)

� x2 � 2x � 2x � 4

� x2 � 4x � 4

14. (x � 4)2 � (x � 4)(x � 4)

� x2 � 4x � 4x � 16

� x2 � 8x � 16

15. (x � 7)2 � (x � 7)(x � 7)

� x2 � 7x � 7x � 49

� x2 � 14x � 49

16. (x � 12)2 � (x � 12)(x � 12)

� x2 � 12x � 12x � 144

� x2 � 24x � 144

17. AB2 � BC2 � AC2

42 � r2 � (2 � r)2

16 � r2 � 4 � 2r � 2r � r2

16 � r2 � 4 � 4r � r2

16 � 4 � 4r

12 � 4r

3 � r

18. AB2 � BC2 � AC2

122 � r2 � (8 � r)2

144 � r2 � 64 � 8r � 8r � r2

144 � r2 � 64 � 16r � r2

144 � 64 � 16r

80 � 16r

5 � r

19. AB2 � AC2 � BC2

212 � r2 � (9 � r)2

441 � r2 � 81 � 9r � 9r � r2

441 � r2 � 81 � 18r � r2

441 � 81 � 18r

360 � 18r

20 � r

20. No; AB2 � AC2 � BC2

152 � 52 � 172

225 � 25 � 289

250 � 289

So, TABC is not a right triangle and AB**** is not perpen-dicular to AC**** . Therefore, AB**** is not tangent to �C.

21. No; AB2 � AC2 � BC2

142 � 52 � 152

196 � 25 � 225

221 � 225

So, TABC is not a right triangle and AB**** is not perpen-dicular to AC**** . Therefore, AB**** is not tangent to �C.

22. AB**** � AD**** , BC**** � DC****

23. aABC � aADC, aBAC � aDAC, aDCA � aBCA

24. T ABC �T ADC

25.

26. Yes. Explanations may vary. Sample answer: JM**** � KM****since tangent segments from the same point are congruent.LJ**** � LK**** since all the radii of a circle are congruent.LM**** � LM**** by the Reflexive Property of Congruence.Then T JLM �T KLM by the SSS Congruence Postulate.

27. Let R represent the center of Earth.

BA2 � AR2 � BR2

BA2 � 39602 � (12,500 � 3,960)2

BA2 � 15,681,600 � 270,931,600

BA2 � 255,250,000

BA � 15,977 miles

BA � BC � 15,977 miles

28. BC2 � FB2 � CF2

39602 � FB2 � 3960.22

15,681,600 � FB2 � 15,683,184.04

FB2 � 1584.04

FB � 39.8

BE2 � CB2 � CE2

BE2 � 39602 � 3960.012

BE2 � 15,681,600 � 15,681,679.2

BE2 � 79.2

BE � 9

FE � BE � FB

� 9 � 39.8

� 49 miles

M

K

J

L





45. m � �x

y

2

2

�

�

x

y

1

1�

� ��1

4�

�

(2�7)

�

� �26

�

� �13

�

Lesson 11.3

11.3 Checkpoint (pp. 602–603)

1. mBCs � 58�, mEFs � 58�; 2. mBCs � 58�, mCDs � 72�;

yes no

3. mCDs � 72�,

mDEs � 360� � (72� � 58� � 100� � 58�)

� 360� � 288�

� 72�;

yes

4. mBFEt � 100� � 58� � 158�,

mCBFt � 58� � 100� � 158�;

yes

5. Arc length of ABr� �13

26

00

�

�� 2π(2) � �

43

�π � 4.19 in.

6. Arc length of DFEt � �13

86

00

�

�� 2π(4) � 4π � 12.57 ft

7. Arc length of MNs � �39600�

�� 2π(6) � 3π � 9.42 cm


1. major arc: BADt or ADBt ;

minor arc: ABr or BDr ;

semicircle: ABDt

2. Sample answer:

3. The measure of an arc is the degree measure of the related central angle (or 360� minus the measure of therelated central angle), while an arc length is a portion ofthe circumference of a circle.

4. mRSr� 60�

5. mRPSt � 360� � 60� � 300�

6. PQRt is a semicircle, so mPQRt � 180�

C

D

B

A

40�

40�


29. D; EF � EG

2x � 3 � 4x � 5

�2x � �8

x � 4

30. G; PR2 � RS2 � PS2

r2 � 362 � (r � 18)2

r2 � 1296 � r2 � 18r � 18r � 324

r2 � 1296 � r2 � 36r � 324

1296 � 36r � 324

972 � 36r

27 � r


31. yes; 5 � 11 14, 5 � 14 11, and 11 � 14 5

32. no; 8 � 14 � 22 23

33. yes; 3 � 13 15, 3 � 15 13, and 13 � 15 3

34. yes; 9 � 18 25, 9 � 25 18, and 18 � 25 9

35. no; 3 � 7 � 10 36. no; 6 � 22 � 28 29

37. x � 2(5) � 10 38. x � �12

�(24) � 12

39. x � �12

�(38) � 19


40. m � �x

y

2

2

�

�

x

y

1

1�

� ��

63�

�

00

�

� ��

63�

� �2

41. m � �x

y

2

2

�

�

x

y

1

1� 42. m � �x

y

2

2

�

�

x

y

1

1�

� �08

�

�

42

� � ��

12�

�

51

�

� ��

64� � �

�

�

43�

� ��23

� � �43

�

43. m � �x

y

2

2

�

�

x

y

1

1� 44. m � �x

y

2

2

�

�

x

y

1

1�

� �7

4�

�

(�03)

� � �4�

�

5(�

�

61)

�

� �140� � ��

151�

� �52

�



7. mQSr� 40� � 60� � 100�

8. mQSPt � 40� � 60� � 120� � 220�

9. mQTRt � mQRr� 40�

10. Arc length of ABr� �34600�

�� 2π(2) � �

49

�π � 1.40 yd

11. Arc length of DEs � �13

06

00

�

�� 2π(6) � �

130�π � 10.47 cm

12. Arc length of FGHt � �360�

3�

60�

140�� 2π(5)

� �23

26

00

�

�� 2π(5)

� �595�π

� 19.20 m


13. PQr; mPQr � 135� 14. DEs; mDEs � 130�

15. LNr; mLNr � 150�

16. ADBt; mADBt � 360� � 75� � 285�

17. WXYt; mWXYt � 360� � 160� � 200�

18. GFHt; mGFHt � 360� � 30� � 330�

19. minor arc 20. minor arc 21. semicircle

22. minor arc 23. major arc 24. semicircle

25. major arc 26. major arc

27. maACB � mABr � 165� 28. maACB � mABr � 90�

29. maACB � mABr � 180� 30. mKLr� 60�

31. mMNs � 55�

32. mLNKt � 360� � 60� � 300�

33. mMKNt � 360� � 55� � 305�

34. NJKt is a semicircle, so mNJKt � 180�

35. maMQL � 180� � (60� � 55�) � 65�

36. mMLr� 65�

37. maJQN � maKQL � 60�

38. mJMr� 60� � 55� � 115�

39. mLNr� 55� � 65� � 120�

40. �32640�� 15� 41. 15� � 6 � 90�

42. If two cities differ by 180� on the wheel, then it is 3:00P.M. in one city when it is 3:00 A.M. in the other city.

43. No; the circles are not congruent.

44. Yes; aACD � aBCE since they are vertical angles;

mADs � mBEs, so ADs � BEs.

45. Yes; UWs and XZs are arcs of congruent circles with thesame measure.

46. No; F is not the center of the circle, so you cannot determine the measures of JKs and GHs.

47. Length of ABr� �34650�

�� 2π(3) � �

34

�π � 2.36 cm


�� 2π(7) � �

73

�π � 7.33 in.

49. Length of ABr� �13

26

00

�

�� 2π(10) � �

230�π � 20.94 ft


�� 2π(4) � �

23

�π � 2.09 cm


�� 2π(12) � 5π � 15.71 m


56

00

�

�� 2π(6) � 5π � 15.71 in.

53. No; they have the same arc length only if the two circlesare congruent circles.

54. Arc length � �03.6406�

�� 2π(5588) � 45 cm


55. C; length of ACs � �180

3�

6�

0�

40�� 2π(8) � 19.6 ft


56. sin 56� � �4x

� 57. sin 34� � �9x

�

4 sin 56� � x 9 sin 34� � x

3.3 � x; 5.0 � x;

cos 56� � �4y

� cos 34� � �9y

�

4 cos 56� � y 9 cos 34� � y

2.2 � y 7.5 � y

58. sin 48� � �1y4�

14 sin 48� � y

10.4 � y;

cos 48� � �1x4�

14 cos 48� � x

9.4 � x


59. �4

200

kmkm� � �

4200

�

�

22

� � �2

100�

60. �752

fint.

� � �5

7�

212

ini.n.

� � �67

02

iinn

.

.� � �

67

02

�

�

11

22

� � �56

�

61. �3

2y7ar

fdts

� � �32�

73ftft

� � �297

fftt

� � �297

�

�

99

� � �13

�

62. �84

lobzs

� � �8 �

41o6zoz

� � �1

428

ozoz

� � �1

428

�

�

44

� � �312�

Quiz 1 (p. 607)

1. tangent 2. secant 3. diameter 4. chord

5. radius 6. point of tangency



7. QP � RP

x � 15

8. QP � RP 9. PQ � PR

x � 1 � 2x � 7 3x � 8 � x

�x � �6 2x � 8

x � 6 x � 4


�� 2π(3) � �

76

�π � 3.67 cm


56

00

�

�� 2π(7) � �

365�π � 18.33 m


�� 2π(10) � �

21

58�π � 4.36 ft

Lesson 11.4


1. JM � HM � 12

2. SR � SN � NR 3. ED � AB

� 15 � 15 x � 4

� 30

4. HG � FG 5. maZWY � maVWU

x � 2 � 5 (x � 10)� � 40�

x � 3 x � 30


1. BE**** is a diameter.

2. PQs and RSr are congruent arcs.

3. PT � RT 4. maMHN � maJHK

x � 1 � 6 (x � 5)� � 50�

x � 7 x � 45

5. DE � AB

x � 8


6. No; AB**** is not perpendicular to CD**** , so AB**** is not a diameter of the circle.

7. No; AB**** does not bisect CD**** , so AB**** is not a diameter ofthe circle.

8. Yes; AB**** is a perpendicular bisector of CD**** , so AB**** is adiameter of the circle.

9. AB � DE 10. maACB � maDCE

x � 7 35� � (x � 6)�

29 � x

11. DB � AB

3x � 1 � x � 7

2x � 6

x � 3

12. AB**** � BC**** (given) and ABs � BCs (If two chords are congruent, then their corresponding minor arcs are congruent.)

13. ABs � CDs (given) and AB**** � CD**** (If two minor arcs are congruent, then their corresponding chords are congruent.)

14. AB**** � CD**** (given) and ABs � CDs (If two chords are congruent, then their corresponding minor arcs are congruent.) aAQB � aCQD by the definition of themeasure of a minor arc.

15. DF � FE � 10 16. mDCs � mBDs � 40�

17. mDCs � mEBs � 110� 18. mBAr� mEDs

mEDCt � mEDs � mDCs (2x � 10)� � (x � 30)�

� 60� � 110� x � 40

� 170�

19. AF � FB 20. maACB � maDCE

x � 6 � 3x � 8 (x � 45)� � 4x�

14 � 2x 45 � 3x

7 � x 15 � x

21. Answers may vary, but the re-creation should be based onthe method shown in Example 2 on page 609.

22. The searcher is constructing a chord of the beacon’s circle and the perpendicular bisector of the chord, whichis a diameter of the circle. By locating the midpoint ofthe diameter, the searcher locates the center of the circle,which is the location of the beacon.


23. a. In a circle, if two chords are congruent, then their corresponding minor arcs are congruent.

b. mADs � mEBs

(15x � 40)� � (10x � 10)�

5x � 50

x � 10

c. mADs � (15x � 40)�

� (15 � 10 � 40)�

� (150 � 40)�

� 110�;

mBEs � mADs � 110�

d. mBDs � 360� � mADs � mBEs � mAEs

� 360�� 110� � 110� � 40�

� 100�


24. mDEs � mABs � 40�

25. mBCs � 180� � mABs � mCDs

� 180� � 40� � 65�

� 75�





26. mAEs � 180� � mDEs 27. mBCDt � mBCs � mCDs

� 180� � 40� � 75� � 65�

� 140� � 140�

28. mABCt � mABs � mBCs 29. mADEt � mADs � mDEs

� 40� � 75� � 180� � 40�

� 115� � 220�


30. �26 �29 31. �12

50� ��

34

� 32. 0.2 � �15

�

Lesson 11.5

11.5 Activity (p. 613)

Step 3. Angle measures will vary, but the measures of allthree angles will be equal.

1, 2. Answers will vary, but in each case,

maRTS � maRUS � maRVS � �12

�maRPS.

3. The measure of an inscribed angle is half the measure ofthe corresponding central angle.


1. maBAC � �12

�mBCs � �12

�(90�) � 45�

2. maDEF � �12

�mDFs � �12

�(160�) � 80�

3. maKMP � �12

�mKNPt

120� � �12

�mKNPt

240� � mKNPt

4. By Theorem 11.8, the triangle is a right triangle. So, x � 90 and y � 90 � 35 � 55.

5. By Theorem 11.8, the triangle is a right triangle.So, y � 90 and 2x � 90 so x � 45.

6. By Theorem 11.8, the triangle is a right triangle.So, y � 90 and x � 90 � 60 � 30.

7. x� � 95� � 180� 8. x� � 90� � 180�

x � 85; x � 90;

y� � 100� � 180� y� � 90� � 180�

y � 80 y � 90

9. x� � 50� � 180�

x � 130;

y� � 80� � 180�

y � 100


1. aA, aB, aC, aD 2. aA and aC, aB and aD

3. maLJK � �12

�mKLs 4. maKJL � �12

�mKMLt

20� � �12

�mKLs 90� � �12

�mKMLt

40� � mKLs 180� � mKMLt

5. maLJK � �12

�mLMKt 6. x� � �12

�(230�)

105� � �12

�mLMKt x � 115

210� � mLMKt

7. 75� � �12

�y� 8. x� � 85� � 180�

150 � y; x � 95;

z� � �12

�(150�) y� � 80� � 180�

z � 75 y � 10


9. maABC � �12

�mACs 10. maABC � �12

�mACs

� �12

�(110�) � �12

�(218�)

� 55� � 109�

11. maABC � �12

�mACs 12. maLMN � �12

�mLNs

� �12

�(180�) � �12

�(68�)

� 90� � 34�

13. maPQR � �12

�mPRs 14. maUTS � �12

�mUSs

� �12

�(134�) � �12

�(238�)

� 67� � 119�

15. maBAC � �12

�mBCs 16. maBAC � �12

�mBCs

32� � �12

�mBCs 78� � �12

�mBCs

64� � mBCs 156� � mBCs

17. maBAC � �12

�mBCs 18. maRST � �12

�mRUTt

114� � �12

�mBCs 120� � �12

�mRUTt

228� � mBCs 240� � mRUTt



19. maPQN � �12

�mPNs 20. maXYZ � �12

�mXWZt

50� � �12

�mPNs 103� � �12

�mXWZt

100� � mPNs 206� � mXWZt

21. maEAB � �12

�mBEs 22. maBDE � �12

�mBEs

47� � �12

�mBEs � �12

�(94�)

94� � mBEs � 47�

23. maAED � maDCE � maBDE � 180�

maAED � 80� � 47� � 180�

maAED � 53�

24. maABD � �12

�mADs

(180� � 47� � 80�) � �12

�mADs

53� � �12

�mADs

106� � mADs

25. maABD � �12

�mADs

� �12

�(106�)

� 53�

26. mDEs � mDAr� mABs � mEBs � 360�

mDEs � 106� � 100� � 94� � 360�

mDEs � 60�

27. Yes; Sample answer: maBAC � 47� � maCDE (fromEx. 22) and maDCE � maACB (vertical angles), so T ABC �T DEC by the AA Similarity Postulate.

28. T ABC is an inscribed triangle and AC**** is a diameter,so T ABC is a right triangle with diameter AC**** ; x � 90;y � 90 � 30 � 60.

29. T KLM is an inscribed triangle and KM**& is a diameter,so T KLM is a right triangle with diameter KM**&; x � 90;y � 90 � 40 � 50.

30. T PQR is an inscribed triangle and PR**** is a diameter,so T PQR is a right triangle with diameter PR**** ; y � 90;x � 90 � 58 � 32.

31. Sample Answer: Position the vertex of the tool on the circle and mark the two points at which the sides intersectthe circle; draw a segment to connect the two points,forming a diameter of the circle. Repeat these steps,placing the vertex at a different point on the circle. Thecenter is the point at which the two diameters intersect.

32. x� � 114� � 180� 33. x� � 100� � 180�

x � 66; x � 80;

y� � 92� � 180� y� � 102� � 180�

y � 88 y � 78

34. x� � 115� � 180�

x � 65;

y� � y� � 180�

2y � 180

y � 90

35. Yes; both pairs of opposite angles are right angles, whichare supplementary angles.

36. Yes; both pairs of opposite angles of an isosceles trapezoidare supplementary.

37. No; if a rhombus is not a square, then the opposite anglesare not supplementary.

38. Yes; both pairs of opposite angles are right angles, whichare supplementary angles.


39. B; maADB � �12

�maACB

� �12

�(80�)

� 40�

40. H; x� � 85� � 180�

x � 95

y� � 100� � 180�

y � 80


41. �5� � �7� � �5� �� 7� � �3�5�

42. �2� � �2� � �2� �� 2� � �4� � 2

43. �6� � �1�4� � �6� �� 1�4� � �8�4� � �4� �� 2�1� � 2�2�1�

44. �8�2��2 � 8�2� � 8�2� � 8 � 8 � �2� �� 2� � 64�4� � 128

45. �3�3��2 � 3�3� � 3�3� � 3 � 3 � �3� �� 3� � 9�9� � 27

46. 2�5� � �1�0� � 2�5� �� 1�0� � 2�5�0� � 2�2�5� �� 2� � 10�2�

47. maB � 90� � 44� � 46�;

sin 44� � �A8B�

AB sin 44� � 8

AB � �sin

844��

AB � 11.5;

tan 44� � �A8C�

AC tan 44� � 8

AC � �tan

844��

AC � 8.3





48. maJ � 90� � 35� � 55�;

sin 35� � �J1K1�

11 � sin 35� � JK

6.3 � JK;

cos 35� � �K11

L�

11 � cos 35� � KL

9.0 � KL

49. maR � 90� � 50� � 40�;

tan 50� � �R5P�

5 � tan 50� � RP

6.0 � RP;

sin 40� � �R5Q�

RQ � sin 40� � 5

RQ � �sin

540��

RQ � 7.8


50. 3x � 5 � 3(2) � 5 51. 8x � 7 � 8(2) � 7

� 6 � 5 � 16 � 7

� 11 � 9

52. x2 � 9 � (2)2 � 9

� 4 � 9

� 13

53. (x � 4)(x � 4) � (2 � 4)(2 � 4)

� (6)(�2)

� �12

54. x2 � 3x � 2 � (2)2 � 3(2) � 2

� 4 � 6 � 2

� 8

55. x3 � x2 � (2)3 � (2)2

� 8 � 4

� 12

Lesson 11.6

11.6 Geo-Activity (p. 620)

Step 2. maAEB � 50�

Step 3. The measure of aAEB is 50� for every student.

Step 4. Answers will vary, but the measure of aAEB will

always be �12

��mABr� mCDs �.

Step 5. The measure of an angle formed by intersectingchords of a circle is equal to half the sum of the measures of the intercepted arcs.


1. x� � �12

��mABr� mDCs � 2. x� � �12

��mABr� mCDs �

x� � �12

�(190� � 70�) x� � �12

�(66� � 70�)

x � �12

�(260) x � �12

�(136)

x � 130 x � 68

3. 72� � �12

��mABr� mCDs �

72� � �12

�(x� � 99�)

72 � �12

�x � 49.5

22.5 � �12

�x

45 � x

4. PN � PL � PM � PK 5. AB � BC � DB � BE

12 � 6 � 12 � x 6 � 6 � x � 9

72 � 12x 36 � 9x

6 � x 4 � x

6. RV � VT � SV � VU 7. FH � HJ � EH � HG

6 � 4 � x � 8 x � 8 � 4 � 10

24 � 8x 8x � 40

3 � x x � 5

8. CG � GE � DG � GF 9. KN � NM � LN � NJ

5 � 5 � 5 � x 16 � 24 � 12 � x

25 � 5x 384 � 12x

5 � x 32 � x


1. Points B and E lie inside the circle.

2. ma1 � �12

��mADs � mBCs � 3. ma1 � �12

��mADs � mBCs �

� �12

�(55� � 65�) � �12

�(88� � 88�)

� �12

�(120�) � �12

�(176�)

� 60� � 88�

4. ma1 � �12

��mBCs � mADs �

� �12

�(110� � 168�)

� �12

�(278�)

� 139�

5. x � 4 � 2 � 6 6. x � 7 � 14 � 8 7. x � 15 � 10 � 18

4x � 12 7x � 112 15x � 180

x � 3 x � 16 x � 12




8. C 9. B 10. D 11. A

12. x� � �12

��mBCs � mADs � 13. x� � �12

��mCDs � mABs �

x� � �12

�(134� � 162�) x� � �12

�(75� � 25�)

x � �12

�(296) x � �12

�(100)

x � 148 x � 50

14. x� � �12

��mABs � mDCs �

x� � �12

�(130� � 96�)

x � �12

�(226)

x � 113

15. 55� � �12

��mCDs � mBAr � 16. 59� � �12

�� mABs � mDCs �

55� � �12

�(89� � x�) 59� � �12

�(70� � x�)

55 � 44.5 � �12

�x 59 � 35 � �12

�x

10.5 � �12

�x 24 � �12

�x

21 � x 48 � x

17. 129� � �12

�� mADs � mBCs �

129� � �12

�(x� � 72�)

129 � �12

�x � 36

93 � �12

�x

186 � x

18. Yes; Sample answer: If two chords intersect and themeasure of each angle formed is the same as the measureof the arc intercepted by the angle, then the circle isdivided into two pairs of congruent arcs since verticalangles are congruent. Suppose one pair of arcs has measure x� and an angle intercepting the arc is a1.

Then ma1 � �12

�(x� � x�) � �12

�(2x�) � x�.

The angle formed by the chords has the same measure asits intercepted arc, so it is a central angle.

19. CN � 10 � 15 � 12 20. x � 9 � 12 � 15

10CN � 180 9x � 180

CN � 18 x � 20

21. x � 4 � 6 � 8 22. x � 10 � 14 � 5

4x � 48 10x � 70

x � 12 x � 7

23. 180� � x� � �12


180� � x� � �12

�(122� � 32�)

180 � x � �12

�(154)

180 � x � 77

103 � x

24. 180� � x� � �12


180� � x� � �12

�(128� � 100�)

180 � x � �12

�(228)

180 � x � 114

66 � x

25. 180� � x� � �12


180� � x� � �12

�(90� � 150�)

180 � x � �12

�(240)

180 � x � 120

60 � x

26. EA � EB � EC � ED; If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.


27. a. BC � 9 � 12 � 3

9BC � 36

BC � 4

b. maACB � �12

��mABs � mDEs�

� �12

�(100� � 80�)

� �12

�(180�)

� 90�

c. mAEs � mABs � mBDs � mDEs � 360�

mAEs � 100� � 36� � 80� � 360�

mAEs � 216� � 360�

mAEs � 144�

d. Yes; Sample answer: by the Vertical Angles Theorem,

aACB � aECD. Also, �AC

CE� � �

192� � �

43

� � �CC

DB�

so the two sides that include the congruent angles areproportional. Thus, T ACB �T ECD by the SASSimilarity Theorem.






28. 32 � b2 � 10229. 162 � 302 � c2

9 � b2 � 100 256 � 900 � c2

b2 � 91 1156 � c2

b � 9.5 34 � c

30. a2 � 32 � 142

a2 � 9 � 196

a2 � 187

a � 13.7


31. ⏐�3⏐ � 3 32. ⏐1⏐ � 1

33. ⏐�19⏐ � 19 34. ⏐50⏐ � 50

35. ⏐2.7⏐ � 2.7 36. ⏐�8⏐ � 8

37. ⏐�10.01⏐ � 10.01 38. ⏐�100⏐ � 100

Quiz 2 (p. 625)

1. 2x � 5 � x

x � 5

2. 3x � 1 � x � 5 3. (3x � 1)� � (x � 63)�

2x � 6 2x � 62

x � 3 x � 31

4. The measure of an inscribed angle is half the measure ofits intercepted arc;

maABC � �12

�mACs

51� � �12

�x�

102 � x

5. x� � maB � �12

�mADs � maC � 58�;

y� � maD � �12

�mBCs � maA � 41�

6. The opposite angles of an inscribed quadrilateral are supplementary;

maB � maD � 180�

x� � 105� � 180�

x � 75;

maA � maC � 180�

98� � y� � 180�

y � 82

7. x� � �12


x� � �12

�(80� � 44�)

x � �12

�(124)

x � 62

8. 135� � �12


135� � �12

�(x� � 107�)

135� � �12

�x � 53.5

81.5 � �12

�x

163 � x

9. x � 14 � 7 � 12

14x � 84

x � 6

11.6 Technology Activity (p. 626)

1. Product will vary, but in each case, JH � JG � JK � JL.

2. The relationship JH � JG � JK � JL is true.

3. no

4. If JL&̂( and JG&̂( are secants of a circle intersecting at a pointJ outside the circle and points K, L, G, and H are points on the circle as shown in the diagram on page 626, thenJH � JG � JK � JL.

5. Measures will vary, but in each case,

maKJH � �12

��mLGs� mKHs �. If 2 secants intersect outside

a circle, then the measure of the angle formed is half thepositive difference of the measures of the intercepted arcs.

Lesson 11.7


1. x2 � y2 � r22. x2 � y2 � r2

x2 � y2 � 22 x2 � y2 � 32

x2 � y2 � 4 x2 � y2 � 9

3. (x � h)2 � (y � k)2 � r2

(x � (�4))2 � (y � (�6))2 � 52

(x � 4)2 � (y � 6)2 � 25

4. y

x

1

1



5.


1. C

2. radius 2, center (0, 0); 3. radius 4, center (2, 0);

(x � h)2 � (y � k)2 � r2 (x � h)2 � (y � k)2 � r2

(x � 0)2 � (y � 0)2 � 22 (x � 2)2 � (y � 0)2 � 42

x2 � y2 � 4 (x � 2)2 � y2 � 16

4. radius 2, center (�2, 2);

(x � h)2 � (y � k)2 � r2

(x � (�2))2 � (y � 2)2 � 22

(x � 2)2 � (y � 2)2 � 4


5. B; radius 2, center (3, 0); 6. A; radius 2, center (0, 0);

(x � 3)2 � y2 � 4 x2 � y2 � 4

7. C; radius 2, center (�3, 0);

(x � 3)2 � y2 � 4



12. radius 5, center (5, 1); 13. radius 6, center (�2, 3);

y

x

2

�2

y

x

2

2

y

x

1

1

y

x

2

�2

y

x

2

2

y

x

2

2

y

x

1

1

14. radius 2, center (2, �5); 15. radius 8, center (0, 5);

16. radius 2, center (�3, 2);

(x � h)2 � (y � k)2 � r2

(x � (�3))2 � (y � 2)2 � 22

(x � 3)2 � (y � 2)2 � 4


(x � h)2 � (y � k)2 � r2 (x � h)2 � (y � k)2 � r2

(x � 0)2 � (y � 1)2 � 22 (x � 3)2 � (y � 3)2 � 1

x2 � (y � 1)2 � 4

19. radius 2.5, center (0.5, 1.5);

(x � h)2 � (y � k)2 � r2

(x � 0.5)2 � (y � 1.5)2 � 2.52

(x � 0.5)2 � (y � 1.5)2 � 6.25


(x � h)2 � (y � k)2 � r2 x2 � y2 � r2

(x � 2)2 � (y � 2)2 � 42 x2 � y2 � 62

(x � 2)2 � (y � 2)2 � 16 x2 � y2 � 36

22. (x � h)2 � (y � k)2 � r2

(x � 0)2 � (y � 0)2 � 102

x2 � y2 � 100

23. (x � h)2 � (y � k)2 � r2

(x � 4)2 � (y � 0)2 � 42

(x � 4)2 � y2 � 16

24. (x � h)2 � (y � k)2 � r2

(x � 3)2 � (y � (�2))2 � 22

(x � 3)2 � (y � 2)2 � 4

25. (x � h)2 � (y � k)2 � r2

(x � (�1))2 � (y � (�3))2 � 62

(x � 1)2 � (y � 3)2 � 36

26. (x � h)2 � (y � k)2 � r2

(x � (�3))2 � (y � 5)2 � 32

(x � 3)2 � (y � 5)2 � 9

27. (x � h)2 � (y � k)2 � r2

(x � 1)2 � (y � 0)2 � 72

(x � 1)2 � y2 � 49

y

x

2

2

y

x�1

1





28. (x � 2)2 � (y � 3)2 � 4

(0 � 2)2 � (0 � 3)2 � 4

(�2)2 � (�3)2 � 4

4 � 9 � 4

13 4

R(0, 0) is outside the circle.

29. (x � 2)2 � (y � 3)2 � 4

(2 � 2)2 � (�4 � 3)2 � 4

02 � (�1)2 � 4

1 4

A(2, �4) is inside the circle.

30. (x � 2)2 � (y � 3)2 � 4

(0 � 2)2 � (�3 � 3)2 � 4

(�2)2 � 02 � 4

4 � 4

X(0, �3) is on the circle.

31. (x � 2)2 � (y � 3)2 � 4

(3 � 2)2 � (�1 � 3)2 � 4

12 � 22 � 4

1 � 4 � 4

5 4

K(3, �1) is outside the circle.

32. (x � 2)2 � (y � 3)2 � 4

(1 � 2)2 � (�4 � 3)2 � 4

(�1)2 � (�1)2 � 4

1 � 1 � 4

2 4

M(1, �4) is inside the circle.

33. (x � 2)2 � (y � 3)2 � 4

(2 � 2)2 � (�5 � 3)2 � 4

02 � (�2)2 � 4

4 � 4

T(2, �5) is on the circle.

34. (x � 2)2 � (y � 3)2 � 4

(2 � 2)2 � (0 � 3)2 � 4

02 � 32 � 4

9 4

D(2, 0) is outside the circle.

35. (x � 2)2 � (y � 3)2 � 4

(2.5 � 2)2 � (�3 � 3)2 � 4

(0.5)2 � 02 � 4

0.25 4

Z(2.5, �3) is inside the circle.

36. circle A: x2 � y2 � 32

x2 � y2 � 9;

circle B: (x � 5)2 � (y � 3)2 � 2.52

(x � 5)2 � (y � 3)2 � 6.25;

circle C: (x � 2)2 � (y � 5)2 � 22

(x � 2)2 � (y � 5)2 � 4

37. Tower A transmits to J;

tower B transmits to K;

towers B and C transmit to L;

no tower transmits to M;

tower C transmits to N.

38. Sample answer: The student did not subtract the coordinates of the center from x and y in the equation and did not square the radius. The equation should be (x � 1)2 � (y � 2)2 � 4.

39. (x � h)2 � (y � k)2 � r240. (x � h)2 � (y � k)2 � r2

(4 � 1)2 � (6 � 2)2 � r2 (5 � 3)2 � (2 � 2)2 � r2

32 � 42 � r2 22 � 02 � r2

9 � 16 � r2 4 � r2

25 � r2 2 � r

5 � r (x � 3)2 � (y � 2)2 � 4

(x � 1)2 � (y � 2)2 � 25


41. D; (x � (�3))2 � (y � 1)2 � 22

(x � 3)2 � (y � 1)2 � 4

42. H; (x � (�3))2 � (y � 0)2 � 52

(x � 3)2 � y2 � 25

(�3 � 3)2 � 02 � 25

02 � 02 � 25


43. P → P�

(1, 3) → (1 � 2, 3) → (3, 3);

Q → Q�

(6, 1) → (6 � 2, 1) → (8, 1);

R → R�

(4, �2) → (4 � 2, �2) → (6, �2);

S → S�

(�1, 0) → (�1 � 2, 0) → (1, 0)



44. P → P�

(1, 3) → (1 � 4, 3 � 1) → (�3, 4);

Q → Q�

(6, 1) → (6 � 4, 1 � 1) → (2, 2);

R → R�

(4, �2) → (4 � 4, �2 � 1) → (0, �1);

S → S�

(�1, 0) → (�1 � 4, 0 � 1) → (�5, 1)

45. P → P�

(1, 3) → (1 � 1, 3 � 1) → (0, 2);

Q → Q�

(6, 1) → (6 � 1, 1 � 1) → (5, 0);

R → R�

(4, �2) → (4 � 1, �2 � 1) → (3, �3);

S → S�

(�1, 0) → (�1 � 1, 0 � 1) → (�2, �1)

46. P → P�

(1, 3) → (1 � 3, 3 � 6) → (4, 9);

Q → Q�

(6, 1) → (6 � 3, 1 � 6) → (9, 7);

R → R�

(4, �2) → (4 � 3, �2 � 6) → (7, 4);

S → S�

(�1, 0) → (�1 � 3, 0 � 6) → (2, 6)

47. reduction;

k � �CC

PP�

� � �14

50� � �

38

�

48. enlargement;

k � �CC

PP�

� � �21

20� � �

151� � 2.2


49. 14 � �3x � 7 50. 11 � x � �2

21 � �3x �x � �13

�7 � x x � 13

51. 20 � 5x � 12 � x

20 � 4x � 12

32 � 4x

8 � x

Lesson 11.8

11.8 Geo-Activity (p. 634)

Step 4. 120�; yes, you can turn the paper clockwise or counterclockwise.

Step 5. rectangle: 180� clockwise or counterclockwise;square: 90� or 180� clockwise or counterclockwise


1. no

2. yes; it can be mapped onto itself by a clockwise or counterclockwise rotation of 180� about its center.

3. yes; it can be mapped onto itself by a clockwise or counterclockwise rotation of 45�, 90�, 135�, or 180�about its center.

4.

A�(0, 0), B�(0, 3), C�(�4, 3)


1. A center of rotation is the fixed point about which a figure is turned when it is rotated.

2. A figure in a plane has rotational symmetry if the figurelooks the same after it is rotated 180� or less.

3. Yes; it can be mapped onto itself by a clockwise or counterclockwise rotation of 180� about its center.


5. no

6. A clockwise rotation of 60� about P maps R onto S.

7. A counterclockwise rotation of 60� about P maps Ronto Q.

8. A clockwise rotation of 120� about Q maps R onto W.

9. A counterclockwise rotation of 180� about P maps Vonto R.


10. Yes; it can be mapped onto itself by a clockwise or counterclockwise rotation of 90� or 180� about its center.


12. no

13. The wheel hub can be mapped onto itself by a clockwiseor counterclockwise rotation of 72� or 144� about its center.

14. The wheel hub can be mapped onto itself by a clockwiseor counterclockwise rotation of 36�, 72�, 108�, 144�, or180� about its center.

15. The wheel hub can be mapped onto itself by a clockwiseor counterclockwise rotation of 45�, 90�, 135�, or 180�about its center.

y

x

1

1A B

C

A�

C� B�





16. 17.

18. 19.

20. 21.

22. CD**** 23. LH**** 24. T FGL

25. TMAB 26. GE****

27.

J(�2, 1) → J�(1, 2)

K(�1, 4) → K�(4, 1)

L(3, 4) → L�(4, �3)

M(3, 1) → M�(1, �3);

The coordinates of the image of the point (x, y) after a90� clockwise rotation about the origin are (y, �x).

28.

D(1, �4) → D�(4, 1)

E(2, 0) → E�(0, 2)

F(5, �2) → F�(2, 5);

The coordinates of the image of the point (x, y) after a90� clockwise rotation about the origin are (�y, x).

x

y

�2E D�

D

E�

F �6

F

x

y

6J

J�

K�

K LM

M� L �

6

Y

W�

Z

P

Z�

Y�

W

XX�

T

SPR�

T�

R S�

B

B�

A

A�

C

C�

D

D�

P

YX X�

Y�

Z �

W

W�

Z

P

S

S�

R

P

R�

T

T�

�

A

B

P

CA�

C�

B�

29.

A(�1, 1) → A�(�1, �1)

B(2, 4) → B�(�4, 2)

C(5, 2) → C�(�2, 5);

The coordinates of the image of the point (x, y) after a90� clockwise rotation about the origin are (�y, x).

30.

X(�2, �3) → X�(2, 3)

O(0, 0) → O�(0, 0)

Z(3, �4) → Z�(�3, 4);

The coordinates of the image of the point (x, y) after a180� rotation about the origin are (�x, �y).

31. The design has rotational symmetry about its center; itcan be mapped onto itself by a clockwise or counter-clockwise rotation of 180�.

32. The design has rotational symmetry about its center; itcan be mapped onto itself by a clockwise or counter-clockwise rotation of 90� or 180�.

33. Yes. The image can be mapped onto itself by a clockwiseor counterclockwise rotation of 180� about its center.

34. Yes; the answer would change to a clockwise or counter-clockwise rotation of 90� or 180� about its center. If youdisregard the colors of the figures, then, for example, thegreen fish in the middle map onto the orange fish and theorange fish map onto the green fish.

35. The center of rotation is the center of the circle.

36. Yes; this piece could be hung upside down because theimage can be mapped onto itself by a clockwise or counterclockwise rotation of 180� about its center.


37. D; in a 90� clockwise rotation,(x, y) → (y, �x).

38. H


39. A � bh � 7 � 13 � 91 ft2

40. A � bh � 9 � 8 � 72 cm2

x

y

4

Z �

Z

X �

O�

6

OX

x

y

2

B �

B

A�

C� 6

CA



41. A � �12

�h(b1 � b2)

� �12

�(10)(6 � 10)

� �12

�(10)(16)

� 80 m2


42. �4�2� � 6.5 43. �9�0� � 3�1�0� � 9.5

44. �2�5�6� � 16 45. �0� � 0

Quiz 3 (p. 639)

1. center (�1, 6), radius 5

2. (x � h)2 � (y � k)2 � r2

(x � 0)2 � (y � (�4))2 � 32

x2 � (y � 4)2 � 9

3. 4.

5. 6.



9. no

10.

A(2, 4) → A�(�2, �4)

B(1, 1) → B�(�1, �1)

C(�1, 3) → C�(1, �3);

The coordinates of the image of the point (x, y) after a180� rotation about the origin are (�x, �y).

x

y

4B �

B

A �

C �

6

CA

y

x

1

1

y

x�1

1

y

x

1

1

y

x

2

2

11.

A(1, 4) → A�(�4, 1)

B(4, 4) → B�(�4, 4)

C(4, 1) → C�(�1, 4)

D(2, 1) → D�(�1, 2);

The coordinates of the image of the point (x, y) after a 90� counterclockwise rotation about the origin are (�y, x).

12.

A(2, 0) → A�(0, �2)

B(4, �1) → B�(�1, �4)

C(3, �3) → C�(�3, �3);

The coordinates of the image of the point (x, y) after a90� clockwise rotation about the origin are (y, �x).

11.8 Technology Activity (p. 640)

1. T A B C is a rotation of T ABC about point P.

2. AP � A P

3. The measure of aAPA is twice the measure of the acuteangle formed at the intersection of lines m and k.

4. The measure of aAPA changes so it is still twice themeasure of the acute angle formed at the intersection oflines m and k.

5. yes

Chapter 11 Summary and Review (pp. 641–645)

1. A secant is a line that intersects a circle in two points.

2. A polygon is inscribed in a circle if all of its vertices lieon the circle.

3. A line in the plane of a circle that intersects the circle inexactly one point is called a tangent.

4. If the endpoints of an arc are the endpoints of a diameter,then the arc is a semicircle.

5. An inscribed angle is an angle whose vertex is on a circleand whose sides contain chords of the circle.

6. A chord is a segment whose endpoints are points on a circle.

x

y

�6 BA �

C�

2

C

A

B �

x

y

2

B � B

A �

C �

D �

8

CD

A





7. A rotation is a transformation in which a figure is turnedabout a fixed point.

8. diameter 9. point of tangency 10. chord

11. center 12. tangent 13. secant

14. AC2 � AB2 � BC215. AC2 � AB2 � BC2

102 � 82 � r2 262 � 242 � r2

100 � 64 � r2 676 � 576 � r2

36 � r2 100 � r2

6 � r 10 � r

16. AC2 � AB2 � BC217. AB � AD

172 � 152 � r2 6 � x

289 � 225 � r2

64 � r2

8 � r

18. AB � AD

2x � 1 � x � 7

x � 6

19. AB � AD

3x � 4 � x � 10

2x � 14

x � 7

20. mDEs � mCEs � mDCs 21. mAEs � 180� � mDEs

� 121� � 59� � 180� � 62�

� 62� � 118�

22. mAECt � mAEs � mCEs

� 118� � 121�

� 239�

23. mBCs � 180� � mDCs � mABr

� 180� � 59� � 36�

� 85�

24. mBDCt � 360� � mBCs 25. mBDAt � 360� � mABr

� 360� � 85� � 360� � 36�

� 275� � 324�

26. Arc length � �35600�

�� 2π(3) � �

56

�π � 2.62 in.

27. Arc length � �13

26

00

�

�� 2π(8) � �

136�π � 16.76 cm


06

00

�

�� 2π(5) � �

295�π � 8.73 m

29. AB � ED

x � 5 � 8

x � 3

30. AB � BD 31. maACB � maDCE

x � 1 � 4x � 5 70� � 2x

6 � 3x 35 � x

2 � x

32. DF � FE

x � 6

33. mBEs � mBDr

x� � 60�

x � 60

34. mBEs � mADs

115� � 55� � 55� � x�

115� � x�

115 � x

35. maABC � �12

�mACs 36. maABC � �12

�mACs

� �12

�(106�) 39� � �12

�mACs

� 53� 78� � mACs

37. maABC � �12

�mADCt

100� � �12

�mADCt

200� � mADCt

38. y � 90; 39. y � 180 � 81 � 99;

x � 90 � 52 � 38 x � 180 � 112 � 68

40. y � 180 � 97 � 83;

x � 180 � 79 � 101

41. x� � �12

��mABs � mCDs � 42. x� � �12

��mABs � mCDs �

x� � �12

�(80� � 110�) x� � �12

�(60� � 30�)

x � �12

�(190) x � �12

�(90)

x � 95 x � 45

43. 180� � x� � �12

��mABs � mCDs �

180� � x� � �12

�(100� � 120�)

180 � x � �12

�(220)

180 � x � 110

70 � x

44. x � 3 � 2 � 6 45. x � 16 � 10 � 8

3x � 12 16x � 80

x � 4 x � 5

46. x � 8 � 10 � 16

8x � 160

x � 20



47. (x � 2)2 � (y � 5)2 � 32

(x � 2)2 � (y � 5)2 � 9

48. (x � (�4))2 � (y � (�1))2 � 42

(x � 4)2 � (y � 1)2 � 16

49. (x � 5)2 � (y � (�2))2 � 72

(x � 5)2 � (y � 2)2 � 49

50. radius 3, center (�4, 1); 51. radius 4, center (2, �3);

52. radius 5, center (0, 0);

53.

54.

55.


ML P

K�

N

N�

L�M�

K

H

H�

P

G

G�

F�

F

BP

B�A

A�

y

x

1

1

y

x

1

1

y

x

1

�1

Chapter 11 Test (p. 646)

1. C is the center. 2. AC2 � AB2 � BC2

D is a point of tangency. 252 � AB2 � 202

AB**** or BF**** is a chord. 625 � AB2 � 400

AB&̂( is a secant 225 � AB2

FC**** or CB**** is a radius. 15 � AB

FB**** is a diameter

3. SR � RV 4. mBCs � 60�

4x � 3 � 2x � 5

2x � 8

x � 4

5. mACs � mABr� mBCs 6. mABDt � 180�

� 50� � 60�

� 110�

7. mCDs � 180� � mACs

� 180� � 110�

� 70�

mCAEt � 360� � mCDs � mDEs

� 360� � 70� � 40�

� 250�


16

50

�

�� 2π(12) � �

233�π � 24.09 in.

9. Arc length � �37650�

�� 2π(5) � �

21

52�π � 6.54 m

10. AB � DC

3x � 1 � 5x � 3

2 � 2x

1 � x

11. x � 9 � 6 � 18

9x � 108

x � 12

12. x� � �12

�(88�)

x � 44

13. x� � 60� � 180�

x � 120;

y� � 105� � 180�

y � 75

14. (x � 6)2 � (y � (�3))2 � 52

(x � 6)2 � (y � 3)2 � 25

15. y

x

1

1





16. 17.

18. The coordinates of the point (x, y) after a 90� counter-clockwise rotation about the origin are (�y, x).

A(0, 3) → A�(�3, 0);

B(3, 1) → B�(�1, 3);

C(�1, �1) → C�(1, �1)

19. (x � 30)2 � (y � 30)2 � 302

(x � 30)2 � (y � 30)2 � 900

Chapter 11 Standardized Test (p. 647)

1. D

2. G; Arc length � �37600�

�� 2π(8) � 9.77 cm

3. A

4. H; x � 3 � 6 � 8

3x � 48

x � 16

5. D; 76� � �12

�x�

152 � x

6. G; ma1 � �12

��mUVs � mSTr �

� �12

�(95� � 67�)

� �12

�(162�)

� 81�

7. B; (x � 7)2 � (y � (�1))2 � 82

(x � 7)2 � (y � 1)2 � 64

8. F; x� � 97� � 180�

x � 83,

y� � 82� � 180�

x � 98

Chapters 1–11 Cumulative Practice (pp. 649–651)

1. AC � AB � BC 2. RT � RS � ST

� 5 � 10 21 � 9 � ST

� 15 12 � ST

3. maSTR � maSTU � maUTR

� 78� � 40�

� 118�

y

x

1

1

y

x

2

2

4. maFGH � maJGF � maJGH

73� � 28� � maJGH

45� � maJGH

5. M � ��x1 �

2

x2�, �y1 �

2

y2�� 3 �

2(�1)�, �

9 �

25

��

22

�, �124��

� (1, 7)

6. ST � �(x�2�� x�1)�2�� (�y2� �� y�1)�2�

� �(��1� �� 3�)2� �� (�5� �� 9�)2�� (��4�)2� �� (��4�)2�� 1�6� �� 1�6�

� �3�2�

� 4�2�

� 5.7

7. 90� � 70� � 20�; 180� � 70� � 110�

8. 90� � 30� � 60�; 180� � 30� � 150�

9. 90� � 48� � 42�; 180� � 48� � 132�

10. 90� � 85� � 5�; 180� � 85� � 95�

11. maAPB � maDPC

(6x � 14)� � 4x�

2x � 14

x � 7;

maAPB � (6(7) � 14)�

� (42 � 14)�

� 28�;

maBPC � 180� � maAPB

� 180� � 28�

� 152�

12.

AB&̂( and CD&̂( do not intersect, and they are not parallel.The two lines are skew lines.

13. Since alternate exterior angles are congruent, you can usethe Alternate Exterior Angles Converse to show l m.

14. Since alternate interior angles are congruent, you can usethe Alternate Interior Angles Converse to show l m.

15. Since same-side interior angles are supplementary, youcan use the Same-side Interior Angles Converse to showl m.

A

B

C D



16. In a plane, if two lines are perpendicular to the same line,then they are parallel to each other.

17. In an isosceles triangle, the base angles are congruent,and by the Triangle Sum Theorem, the sum of the anglesof a triangle is 180�.

180� � 124� � 2x�

56 � 2x

x � 28

So, the measure of each base angle is 28�.

18. AB2 � BC2 � AC2

82 � 62 � 122

64 � 36 � 144

100 144

obtuse

19. BC**** is the shortest side, so aA is the smallest angle;AC**** is the longest side, so aB is the largest angle.

20. Yes; aA � aJ, aC � aL and the included sides, AC****and JL**** are congruent. So, T ABC �T JKL by the ASACongruence Postulate.

21. No; you only know that GF**** � DE**** and GE**** � GE**** bythe Reflexive Property of Congruence.

22. Yes; Sample answer: PQ**** � RQ**** and PS**** � RS**** andtheir included angles, aP and aR, are congruent. So,T PQS �T RQS by the SAS Congruence Postulate.

23. Yes; hypotenuses XU**& and VU**** are congruent and legs XY**** and VW**** are congruent in right triangles T XYU andT VWU. So, T XYU �T VWU by the HL CongruenceTheorem.

24. AB � AD 25. x� � 63� � 180�

2x � x � 8 x � 117;

x � 8; y� � x�

y� � 90� y � 117

y � 90

26. maK � maL � maM � maN � 360�

85� � 120� � x� � 88� � 360�

x � 293 � 360

x � 67

27. x� � 70� � 180�

x � 110;

y� � 110� � 180�

y � 70;

z� � 70� � 180�

z � 110

28.

2(2x) � 2(3x) � 60

4x � 6x � 60

10x � 60

x � 6

length � 3x � 3(6) � 18 cm

width � 2x � 2(6) � 12 cm

29. Sample answer: aCBD � aCAE and aBCD � aACE(Reflexive Property of Congruence). Therefore,T BCD �T ACE by the AA Similarity Postulate.

30. BD � �12

�AE

5 � �12

�x

10 � x

31. A � bh � bh

� (16)(18) � (20 � 16)(6)

� (16)(18) � (4)(6)

� 288 � 24

� 312 m2

32. A � �12

�h(b1 � b2)

� �12

�(8)(12 � 19)

� �12

�(8)(31)

� 124 ft2

33. A � �12

�d1 d2

� �12

�(16)(10)

� 80 cm2

34. A � πr235. A � bh

� π(18)2 84 � 14h

� 324π 6 m � h

� 1018 yd2

2x

3x





36. S � 2B � Ph

� 2(6 � 8) � (2 � 6 � 2 � 8)(4)

� 2(48) � (12 � 16)(4)

� 96 � (28)(4)

� 96 � 112

� 208 in.2;

V � Bh

� (6 � 8)(4)

� (48)(4)

� 192 in.3

37. S � πr2 � πrl 38. S � 2πr2 � 2πrh

� π(8)2 � π(8)(17) � 2π(6)2 � 2π(6)(10)

� 64π � 136π � 72π � 120π

� 200π � 192π

� 628 cm2; � 603 m2;

82 � h2 � 172 V � πr2h

64 � h2 � 289 � π(6)2(10)

h2 � 225 � 360π

h � 15; � 1131 m3

V � �13

�πr2h

� �13

�π(8)2(15)

� �13

�π(64)(15)

� 320π

� 1005 cm3

39. S � 4πr2

� 4π(12)2

� 576π

� 1810 mm2;

V � �43

�πr3

� �43

�π(12)3

� �43

�π(1728)

� 2304π

� 7238 mm3

40. sin R ��leg

h

o

y

p

p

p

o

o

te

s

n

it

u

e

s

ae

R��

13

59� � 0.3846

cos R ��leg

h

a

y

d

p

j

o

a

t

c

e

e

n

n

u

t

s

ae

R��

33

69� � 0.9231

tan R ��l

l

e

e

g

g

o

ad

p

j

p

a

o

c

s

e

i

n

te

t aa

R

R��

13

56� � 0.4167

41. Since sin R � 0.3846, maR � sin�1 0.3846 � 22.6�;

maT � 90� � maR � 90� � 22.6� � 67.4�

42. x� � 30� � 90�

x � 60;

tan 30� � �7a

�

a tan 30� � 7

a � �tan

730��

a � 12.1;

sin 30� � �7c

�

c sin 30� � 7

c � �sin

730��

c � 14

43. x� � 65� � 90�

x � 25;

sin 65� � �8r

�

8 sin 65� � r

7.3 � r;

cos 65� � �p

8�

8 cos 65� � p

3.4 � p

44. m2 � 92 � 162

m2 � 81 � 256

m2 � 377

m � 18.4;

x� � tan�1 �196� � tan�1 0.5625 � 29.4�;

y� � 90� � 29.4� � 60.6�

45. d2 � 202 � 292

d2 � 400 � 841

d2 � 441

d � 21;

y� � sin�1 �22

09� � sin�1 0.6897 � 43.6�;

x� � 90� � 43.6� � 46.4�

46. A � �m36

A0Br�

� � πr2

� �37650�

�� π(6)2

� 7.5π

� 23.6 ft2

47. mABr� 75�;

mACBt � 360� � 75� � 285�



48. Arc length � �37650�

�� 2π(6) � 2.5π � 7.9 ft

49. (3x � 30)� � (2x � 7)�

x � 30 � 7

x � 37

50. x� � �12

�(92�)

x � 46

51. x� � �12

�(184� � 106�)

x � �12

�(290)

x � 145

52. 3 � x � 9 � 4

3x � 36

x � 12

53. (x � 0)2 � (y � (�3))2 � 72

x2 � (y � 3)2 � 49

54. 55.

56.

A�(�2, �5), B�(�4, �1),

C�(�1, �2)

x

y

2B �

B

A �C �

2C

A

y

x

1

1

C� C

B� B

A� Ay

x

1

1

CB

A

C� B�

A�



chapter 11 · chapter 11 worked-out solution key. chapter 11continued 188 geometry, concepts and...

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