Multiple Choice 17 - Mrs. Lawson's Math...

Chapter 12 Test RETAKE REVIEW

Multiple Choice

Identify the choice that best completes the statement or answers the question.

Find the value of x. If necessary, round your

answer to the nearest tenth. The figure is not

drawn to scale.

1.

17

x

5

a. 19.34

b. 10.49

c. 110

d. 9.22

2. The circles are congruent. What can you conclude from the diagram?

C FO

AP

E

B

D

)

)

a. arc CAB arc FDE c. arc AB arc DE

b. arc DF arc AC d. none of these

Use the diagram. is a diameter, and

. The figure is not drawn to scale. C

A

D

BP

3. Which statement is NOT true?

a. arc AC arc BD

b. c. d.

4. Find the measures of the indicated angles. Which

statement is NOT true? (The figure is not drawn to

scale.)

O

a bc

d

a. a = 53°

b. b = 106°

c. c = 73°

d. d = 37°

5. Compare the quantity in Column A with the

quantity in Column B.

Z

P

QX

A

Y

3

26 4

Column A Column B

AP AX

a. The quantity in Column A is greater.

b. The quantity in Column B is greater.

c. The two quantities are equal.

d. The relationship cannot be determined from the

information given.

6. A low-wattage radio station can be heard only

within a certain distance from the station. On the

graph below, the circular region represents that part

of the city where the station can be heard, and the

center of the circle represents the location of the

station. Which equation represents the boundary for

the region where the station can be heard?

O 4 8–4–8 x

4

8

–4

–8

y

a. (x – 6) + (y – 1) = 32

b. (x + 6) + (y + 1) = 32

c. (x – 6) + (y – 1) = 16

d. (x + 6) + (y + 1) = 16

7. Write the standard equation of the circle in the

graph.

O 4 8–4–8 x

4

8

–4

–8

y

a. (x + 3) + (y – 2) = 9

b. (x – 3) + (y + 2) = 9

c. (x – 3) + (y + 2) = 18

d. (x + 3) + (y – 2) = 18

Short Answer

Assume that lines that appear to be tangent are

tangent. O is the center of the circle. Find the

value of x. (Figures are not drawn to scale.)

8. 111

O x°

9. 12

O

x°

QP

10. is tangent to circle A at B and to circle D at C (not drawn to scale).

AB = 7, BC = 18, and DC = 5. Find AD to the nearest tenth.

A

B

C

D

11. is tangent to circle O at B. Find the length of the radius r for AB = 5 and AO = 8.6. Round to the nearest tenth if

necessary. The diagram is not to scale.

B

O

A

r

12. A chain fits tightly around two gears as shown. The distance between the centers of the gears is 20 inches. The

radius of the larger gear is 11 inches. Find the radius of the smaller gear. Round your answer to the nearest tenth, if

necessary. The diagram is not to scale.

19 inches

13. and are all tangent to O (not drawn to

scale). JA = 9, AL = 10, and

CK = 14. Find the perimeter of .

L

J

K

O

C

A B

14. Pentagon RSTUV is circumscribed about a circle.

Solve for x for RS = 10, ST = 13,

TU = 11, UV = 12, and VR = 12. The figure is not

drawn to scale.

R

S

TU

V

x

In the figure, and are tangent to circle

O and bisects . The figure is not

drawn to scale.

O

P

A

B

C

D

15. For = 46, find .

16. For = 50, find .

Find the value of x. If necessary, round your

answer to the nearest tenth. The figure is not

drawn to scale.

17. , , FG = 40, RS = 37, OP = 19

F

G

R

S

OP

Q

x

18. , , , MN = 6 feet

M

O

R

A

P

N

x

19. D

C

BA

P

x°

20.

6

8

x

21.

x

23

7

22.

10

8

15x

23. The figure consists of a chord, a secant and a tangent to the circle. Round to the nearest hundredth, if necessary.

4

79

15x

24. AB = 20, BC = 6, and CD = 8

C

A

B

x

D

25. and are diameters. Find the measure of arc

ZWX. (The figure is not drawn to scale.)

W

Z

X RC

Y

46

95

Use the diagram. is a diameter, and

. The figure is not drawn to scale.

C

A

D

BP

26. Find for = 66.

27. Find m(arc BD) for m(arc AC) = 43.

28. The radius of circle O is 18, and OC = 13. Find AB.

Round to the nearest tenth, if necessary. (The figure

is not drawn to scale.)

A B

O

C

29. m R = 22. Find m O. (The figure is not drawn to

scale.)

O

N

Q

R

30. Given that DAB and DCB are right angles and

m BDC = 41, what is the measure of arc CAD?

(The figure is not drawn to scale.)

A

D

C

B

31. Find the measure of BAC. (The figure is not

drawn to scale.)

A

C

O

B

32. Find x. (The figure is not drawn to scale.)

A

C

O

B 46

x

33. Find m BAC. (The figure is not drawn to scale.)

A

C

O

B

34. is tangent to circle O at C, m(arc AEC) = 279,

and m ACE = 85. Find m DCE.


C

A

E

O

B

D

35. If m(arc BY) = 40, what is m YAC? (The figure is

not drawn to scale.)

B

A

Y

O

C

36. In the circle, m(arc AD) = 94, and m D = 76. Find

m DCQ.


A

D

C

B

P

Q

37. m(arc DE) = 96 and m(arc BC) = 67. Find m A. (The figure is not drawn to scale.)

D

A

E

C

B

38. Find the value of x for m(arc AB) = 46 and m(arc

CD) = 25. (The figure is not drawn to scale.)

C

A

D

B

O

x

39. Find m D for m B = 50. (The figure is not drawn

to scale.)

A

D

C

B

40. m S = 36, m(arc RS) = 118, and is tangent to

the circle at R. Find m U.


R

UST

41. A park maintenance person stands 16 m from a

circular monument. If you assume her lines of sight

form tangents to the monument and make an angle

of 34°, what is the measure of the arc of the

monument that her lines of sight intersect?

42. Find the diameter of the circle for BC = 16 and DC

= 28. Round to the nearest tenth.

(The diagram is not drawn to scale.)

A

DC

OB

43. Find AB. Round to the nearest tenth if necessary.

C

DA

B

8

15

44. A footbridge is in the shape of an arc of a circle.

The bridge is 7 ft tall and 16 ft wide. What is the

radius of the circle that contains the bridge? Round

to the nearest tenth.

Write the standard equation for the circle.

45. center (2, 7), r = 4

46. center (–6, –8), that passes through (0, 0)

47. The center of a circle is (h, 7) and the radius is 10.

The circle passes through (3, –1). Find all possible

values of h.

48. Find the center and radius of the circle with

equation (x + 9) + (y + 5) = 64.

49. Determine whether a tangent line is shown in the

diagram, for AB = 7, OB = 3.75, and AO = 8.

Explain your reasoning. (The figure is not drawn to

scale.)

B

O

A

50. In circle O, and are tangents. (The figure is not drawn to scale.)

A

P

B

OO

C

D

a. Prove that .

b. Find m BOD for m AOP = 64. Explain your reasoning.

51. In , NL = NM, and the perimeter is 46 cm. A, B, and C are points of tangency to the circle. MC = 4 cm. Find

NL. Explain your reasoning. (The figure is not drawn to scale.)

N

M

L

B

C

A

52. CD = 42, OM = 17, ON = 16, , (The figure is not drawn to scale.)

C

D

E

F

OM

N

a. Find the radius. If your answer is not an integer, express it in radical form.

b. Find FN. If your answer is not an integer, express it in radical form.

c. Find EF. Express it as a decimal rounded to the nearest tenth.

53. a. Find x. (The figure is not drawn to scale.)

b. Is the triangle equilateral, isosceles, or scalene?

Explain.

P

Q

R

(8x – 10)°(6x)°

(10x + 10)°

54. Given: m X = 150, , m Y = 92. Find

each measurement. (The figure is not drawn to

scale.)

Y

X

W

Z

a. m Z

b. m(arc WZ)

c. m W

d. m(arc WX)

55. Given: arc AB arc CD; is tangent to circle P

at B. Explain why .


A B

D X

P

C

56. Given: arc CF = arc DE

Prove:

F

C

D

E

O

57. m A = 20 and m(arc BC) = 88 (The figure is not

drawn to scale.)

C

BA

y°

x°

D

)

a. Find x.

b. Find y.

58. The diameter of a circle has endpoints P(–10, –8) and Q(4, 4).

a. Find the center of the circle.

b. Find the radius. If your answer is not an integer, express it in radical form.

c. Write an equation for the circle.

59. Graph the circle with equation (x + 1) + (y – 3) =

9.

60. The equation (x + 5) + (y + 3) = 169 models the

position and range of the source of a radio signal.

Describe the position of the source and the range of

the signals.

Essay

61. These two noncongruent circles intersect in exactly one point. Their common tangent is 24 cm, and the distance

between their centers is 25 cm.

A B

D

C24 cm

r

a. What is the sum of the two radii?

b. Express the radius of circle B in terms of r, the radius of circle A.

c. Find the radius of circle B. Show your work.

62. The chord-tangent theorem states that the measure of the angle formed by a chord and a tangent segment that

intersect at the tangent’s point of contact is equal to one half the measure of the intercepted arc. Below are three

possible cases of this theorem.

Case I Case II Case III

A

B

O X

C

A

BY

O X

C

A

BY

O X

C

Given that Case I is true, prove Case II of the chord-tangent theorem.

63. Given: The circles share the same center, O, m MON = 120, and

m(arc AX) = m(arc BY) = 106.

B

P

A

N

O

M

Y

X

a b120°

106°

106°

a. Find m P. Show your work.

b. Find a and b. Explain your reasoning.

Other

64. If a circle is inscribed in a square, then the sides of

the square are tangent to the circle. If the circle is

circumscribed about the square, are the sides of the

square tangent to the circle? Use a diagram to

explain your reasoning.

65. Show that it is not possible for the lengths of the

segments of two intersecting chords to be four

consecutive integers.

m + 3

+ 1

+ 2

m

m

m

66. Explain why arc AC arc BD, given .

A B

C D

67. If ABCD is a rectangle inscribed in circle O, do

both diagonals contain the center of the circle?

Explain.

A B

D C

O


28. ANS:

24.9









29. ANS:

44

PTS: 1 DIF: L1 REF: 11-3 Inscribed Angles








30. ANS:

262









31. ANS:

28.5









32. ANS:

23







TOP: 11-5 Example 2 KEY: equation of a circle | center | radius | point on the circle



47. ANS:

9, –3





KEY: equation of a circle | center | radius | point on the circle | algebra



48. ANS:

center (–9, –5); r = 8





TOP: 11-5 Example 3 KEY: center | circle | coordinate plane | radius



49. ANS:

No, because .






TOP: 11-1 Example 3

KEY: tangent to a circle | right triangle | Pythagorean Theorem | properties of tangents



TV.LV20.52

50. ANS:

a. Since all radii are congruent, . Tangent segments from a point outside a circle

to two points on the circle are congruent, so . By the Reflexive Property,

. It follows that by SSS.

b. 116; since , then by CPCTC, so m BOP = 64.

BOD and BOP form a linear pair, so m BOD = 180 – 64 = 116.






TOP: 11-1 Example 3

KEY: multi-part question | tangent to a circle | right triangle | proof | properties of tangents | reasoning



TV.LV20.52

51. ANS:

and, by the Tangent Theorem, NC = NA. By subtraction, . Also by the Tangent Theorem,

and , so . The perimeter is 46 cm, so

. By substitution, , so . Since

, , or 19 cm.






TOP: 11-1 Example 5

KEY: reasoning | tangent to a circle | tangent | properties of tangents | Tangent Theorem



TV.LV20.52

52. ANS:

a.

b.

c. 43.5






KEY: multi-part question | chord | radius | congruent chords



53. ANS:

a. 15

b. Scalene; the arc measures are 110°, 90°, and 160°. Since the arcs are not congruent,

neither are the chords that intercept them.





STA: FL MA.A.2.4.2 | FL MA.B.2.4.1 | FL MA.C.1.4.1

KEY: chord | chord-arc relationship | multi-part question | reasoning



54. ANS:

a. 30

b. 150

c. 88

d. 34






KEY: chord | inscribed angle-arc relationship | circle



55. ANS:

Since and are inscribed in semicircles, they are both right angles. Also, is a right angle

because it forms a linear pair with . All right angles are congruent, so .






KEY: tangent | properties of tangents | chord | inscribed angle-arc relationship | circle | diagonal | rectangle | angle

measure | arc | circle | inscribed angle | intercepted arc | reasoning



56. ANS:

Statements Reasons

1. 1. Inscribed angles intersecting the same arc are .

2. arc CF arc DE 2. Given

3. (arc CF)

(arc DE)

3. Inscribed Angle Theorem

4. (arc CF)

(arc CF)

4. Substitution

5. 5. Substitution

6. 6. Definition of congruence

7. 7. Reflexive property

8. 8. AAS





TOP: 11-5 Example 3 KEY: circle | coordinate plane | center



60. ANS:

The signal is located at (–5, –3) and has a range of 13 units.





TOP: 11-5 Example 4 KEY: circle | coordinate plane | center | word problem



ESSAY

61. ANS:

[4] a. 25 cm

b. 25 – r

c. Let r + x represent the radius of circle B.

x + 24 = 25 ; x = 7; r + 7 = 25 – r; r = 9

r + x = 9 + 7 = 16; 16 cm

[3] one computational error

[2] incomplete work in part c OR correct answers with no work shown

[1] only two parts correct






KEY: extended response | rubric-based question | point of tangency | properties of tangents | Pythagorean Theorem

| right triangle | tangent to a circle



TV.LV20.52

62. ANS:

[4] By the inscribed angle theorem, . By Case I, we know

.

It follows that .

Therefore, .

[3] error in angle or arc name

[2] illogical sequence of steps OR no reasons stated for important steps

[1] missing step






KEY: extended response | rubric-based question | circle | angle measure | angle-arc relationship | arc addition | arc

measure | diameter | inscribed angle | inscribed angle-arc relationship | intercepted arc | proof | tangent-chord angle



63. ANS:

[4] a. 60; 120 + 90 + 90 + m P = 360

b. ; a – b = 120

a + b + 2(106) = 360; a + b = 148

a = 134; b = 14

[3] one computational error

[2] correct answers with no work shown

[1] only one part correct






KEY: extended response | rubric-based question | circle | angle measure | angle-arc relationship | arc addition | arc

measure | inscribed angle | inscribed angle-arc relationship | intercepted arc | tangent to a circle | central angle |

intersection on the circle | intersection outside the circle



OTHER

64. ANS:

No; a tangent to a circle is a line that intersects the circle in exactly one point. Each segment of the square

intersects the circle in exactly two points. In the diagram, for example, intersects the circle in two points, A

and B, so is not tangent to the circle.

A

BD

C






KEY: reasoning | tangent to a circle | inscribe | circumscribe | square | tangent | properties of tangents



TV.LV20.52

65. ANS:

Let m, m + 1, m + 2, and m + 3 represent the four consecutive numbers. Then the product of the greatest and least

numbers will equal the product of the two consecutive middle numbers. Solving the equation m(m + 3) = (m + 1)(m

+ 2) for m results in m2 + 3m = m2 + 3m + 2, or 0 = 2, which is false.






KEY: circle | intersection inside the circle | segment length | algebra | proof | reasoning



66. ANS:

Draw . Then because alternate interior angles of parallel lines are congruent. Thus, arc AC

arc BD, since congruent angles intercept congruent arcs in a circle or in congruent circles.






KEY: chord | angle-arc relationship | reasoning



67. ANS:

Multiple Choice 17 - Mrs. Lawson's Math...

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