1© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

LeSSon 24-1 1. a.

� ��CF

b. DE DB DF, , or

c. AB BEor

d. DF

2. C

3. 17 units

4. 12 units

5. 4

LeSSon 24-2 6. a. 26.5 cm

b. 18 cm

7. a. 23.4 cm

b. 22.7 cm

8. a. Both are segments whose endpoints are points on the circle.

b. Sample answer. A diameter must contain the center of the circle, but a chord doesn't have to contain the center.

9. B

10. a. 4

b. 4

c. mnp

d. 15

LeSSon 24-3 11. a. 9

b. 16

c. AN 5 1 2a b b( )2 2 or AN 5 1a ab22

d. 10

12. a. 38

b. 2 5

13. C

14. a. ST > TQ because radii of the same circle are congruent, and RS > RQ because tangents to a circle from an external point are congruent.

b. kite. It has two pairs of congruent, adjacent sides.

c. They are congruent because each is a right angle.

d. They are supplementary because the sum of all 4 angles is 360° and the other two angles are right angles.

e. TR is the perpendicular bisector of SQ.

15. a. 39

b. 8.1

LeSSon 25-1 16. a. 150°

b. 30°

c. any three of arcs �BAD, �BAC, �ABC, �ABD

d. �AD, �DC, �CB

17. a. 38°

b. 218°

c. 322°

d. 142°

e. 38°

18. B

19. a. 11

b. m∠APC 5 132°, m∠CPB 5 48°

20. a. 144°

b. 72°

c. 144°

d. 216°

Answers to Geometry Unit 4 Practice

2© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

LeSSon 25-2 21. a. 56°

b. 90°

c. 45°

d. 34°

e. 73°

22. a. 20°

b. 27.5°

c. 25.5°

d. 146°

e. 214°

23. B

24. a. rectangle

b. 60°

c. 120°

d. Sample answer. The two angles are inscribed angles that intercept the same arc, PS

�. So they

have the same measure.

25. a. 42°

b. 42°

c. 19°

d. 38°

e. 40°

LeSSon 25-3 26. a. 9

b. m�XW 5 53°, m�ZX 5 109°, m�ZY 5 63°

c. 58°

d. 10

27. a. 70°

b. 79°

c. 149°

d. 30.5°

28. a. 90°

b. 55°

c. 60°

d. No. ∠QED and ∠QDE can be named as ∠CED and ∠EDB, respectively. Those two angles intercept the unequal arcs of 60° and 65°, so ∠QED is not congruent to ∠QDE and so QE QD.

29. D

30. a. never

b. sometimes

c. always

d. always

LeSSon 25-4 31. a. m�ADB 5 280°, m∠T 5 100°

b. m�AB 5 150°, m∠T 5 30°

c. m�ADB 5 230°, m�AB 5 130°

d. m∠T 5 m�AB 5 90°, m�ADB 5 270°

32. a. 17

b. 50°

c. 88°

d. 19°

33. a. 33°

b. 79°

c. 226°

d. 134°

e. 46°

34. a. m�CD 5 170°, m∠A 5 15°

b. m�BD 5 194°, m�CD 5 74°

c. m�BD 5 144°, m∠A 5 31.5°

d. m�BC 5 78°, m�BD 5 162°

35. B

3© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

LeSSon 26-1 36. B

37. a. (x2, y1). Points on a horizontal line have the same y value.

b. 1

x xy

2,1 2

1

c. MT 5 2x x( )4

2 12

5 2x x12

( )2 12 ,

TN 5 2x x( )4

1 22

5 2x x12

( )1 22

d. 2x x( )2 12

e. Yes. Sample answer. (x1 2 x2)2 5 (x2 2 x1)

2,

so 2x x12

( )2 12 5 2x x

12

( )1 22 and MT 5 TN.

38. a. (a 1 c, b 1 d)

b. (0, 0)

c. 1 1

p t q r2

,2

d. (2a 1 4b, 4a 1 3b)

39. a. (0, 0)

b. (0, 0),

a b2,2

, (a, b),

a b32, 32

, (2a, 2b)

40. a. (2m, 0), (2m, 2n), (0, 2n)

b. 1m n2 2

c. (4m 1 4n) units

d. 4mn units2

LeSSon 26-2 41. D

42. B

43. a. If two parallel lines are cut by a transversal, corresponding angles are congruent.

b. Reflexive Property of Congruence

c. It represents the slope of � ���AC.

d. It represents the slope of � ��DE.

e. AFC > DEF by the AA Similarity Postulate.

5AFFC

DFFE

because corresponding sides of

similar triangles are proportional. Since AFFC

and DFFE

represent the slopes of � ���AC and

� ��DE ,

respectively, then � ���AC and

� ��DE have the same

slope.

44. a. DCB > EAC by SAS

b. They are complementary.

c. 90°. Sample answer. m∠2 1 m∠BCA 1 m∠3 5 180 and m∠2 1 m∠3 5 90. Subtracting, m∠BCA 5 90.

d. Line m contains (a, b) and (0, 0) so the slope is ba

ba

00

2

25 . Line n contains (2b, a) and (0, 0)

so the slope is ab

ab

00

2

2 25

2.

e. ⋅ba

ab

abab

12

52

52

45. a. Line q is vertical.

b. Line q has a positive slope.

c. Line q goes down from left to right.

d. Line q has a slope of 0 (or is horizontal).

e. Line q goes up from left to right.

LeSSon 26-3

46. a. 1 1

X

0 82

, 6 02

5 X(4, 3)

b. 2 1 1

Y

4 02

, 0 62

5 Y(22, 3)

c.

Z 4 8

2, 0 0

22 1 1 5 Z(2, 0)

d. Sample answer. The effect of using even integers is that there are no fractions or decimals in the ordered pairs for the midpoints.

4© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

47. a. Slope of AX : 2

2 25

3 04 ( 4)

38

;

Slope of BY : 3 02 8

310

2

2 25 2 ;

Slope of CZ : 2

25

25 2

6 00 2

62

3

b. � ���AX : y 2 0 5

38

(x 2 (24)) or y 5 38

(x 1 4)

� ���BY : y 2 0 5 2

310

(x 2 8)

� ��CZ : y 2 6 5 23(x 2 0) or y 2 6 5 23x

48. a. 38

(x 1 4) 5 2310

(x 2 8)

b. 40

x38( 4)1 5 40

x310

( 8)2 2

15(x 1 4) 5 212(x 2 8)

15x 1 60 5 212x 1 96

27x 5 36

x 5 3627

5 43

c. Sample answer. Using y 5 38

(x 1 4), the

equation for � ���AX , solve for y:

y 5 38

(x 1 4)

5 1

38

43

123

5

38

163

5 2

d. (x, y) 5

43, 2

49. a. Sample answer. An equation for � ��CZ is

y 2 6 5 23x. Check that the ordered pair

43, 2

satisfies that equation:

y 2 6 5 23x

2 2 6 5 2

3 4

3 24 5 24 It checks.

b. Sample answer. In Item 46, I found the three midpoints of the sides, and in Item 47, I found the equations for the three medians. In Item 48, I found the point of intersection for two of the medians, and in Item 49, I showed that the third median intersects the other two medians at that same point. In all, I showed that the medians for a particular triangle are concurrent.

50. C

LeSSon 26-4 51. a. (15, 21)

b. (11, 19)

c. (4, 22)

d. (0.5, 213.5)

52. a. slope of AB : 526

13

; slope of AP : 13

;

AP : AB 5 10 : 40 5 1 : 2

b. slope of AB : 7 9

5 ( 3)168

2 2

2 25

2 5 22;

slope of AP : 3 9

3 ( 3)126

2 2

2 25

2 5 22;

AP : AB 5 6 5 : 8 5 5 3 : 4

53. C

54. (2.2, 3.1)

55. (1, 4)

LeSSon 27-1 56. a. (21, 11), (21, 5)

b. (28, 4), (22, 4)

57. a. (x 2 5)2 1 y2 5 121

b. (x 2 2)2 1 (y 1 3)2 5 36

c. x2 1 (y 1 4)2 5 16

d. (x 2 2.5)2 1 (y 2 6)2 5 72.25?

5© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

58. a. center (3, 2), radius 1

b. center (27, 211), radius 10

c. center (3.8, 1.2), radius 5

d. center (a, 2b), radius m

59. a. x 5 4

b. y 5 3

c. (4, 3)

d. TA 5 1 52 2 2 22 2 ; TB 5 1 52 2 2 22 2 ;

TC 5 1 52 2 2 22 2

e. (x 2 4)2 1 (y 2 3)2 5 8

60. B

LeSSon 27-2 61. a. 1 9; (x 1 3)2

b. 1 81; (x 2 9)2

c. 1 254

; (y 1 52

)2

d. 1 56.25; (y 1 7.5)2

e. 1 a2; (x 1 a)2

62. a. (x 1 3)2 1 (y 1 2)2 5 13; I added 13 to each side.

b. (x 2 2)2 1 (y 1 5)2 5 29; I added 29 to each side.

c. (x 1 1)2 1 (y 2 2.5)2 5 8.25; I added 7.25 to each side.

d. (x 2 4.5)2 1 (y 2 0)2 5 25.25; I added 20.25 to each side.

63. a. center (25, 26); radius 39

b. center (24, 1); radius 5

c. center (10, 3); radius 10

d. center 2

12, 12

; radius 1

64. a. Yes. The center is (23, 3.5).

b. not a circle

c. Yes. The center is (0, 21.5).

d. not a circle

65. B

LeSSon 28-1 66. B

67. a. 5yx20

2

b. 5yx12

2

c. 52yx8

2

d. 5yx24

2

68. a. −yy28

2

5

b. 52yx4

2

c. y 5 x2

d. 5yy10

2

69. yx5

2

52

70. a. (5, 0)

b. (1.25, 0)

c. (0, 22.5)

d. r4

, 0

LeSSon 28-2 71. a. (3, 5)

b. (25, 0)

c. (3, 22)

d. (2a, b)

72. a. Sample answer. The vertex is halfway between the focus and the closest point on the directrix, so the vertex is (21, 2).

b. Sample answer. p is the distance between the vertex and the directrix, so p 5 3.

c. The parabola opens upward. Sample answer. The directrix for this parabola is a horizontal line and the focus and vertex are above that line, so the parabola opens upward.

6© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

d. y 2 k 5 p14

(x 2 h)2; (h, k) 5 (21, 2), p 5 3

e. y 2 2 5 112

(x 1 1)2

73. a. (i) opens to the right;

(ii) x 2 h 5 p14 (y 2 k)2;

(iii) (h, k) 5 (23, 1), p 5 3;

(iv) x 1 3 5 112(y 2 1)2

b. (i) opens down;

(ii) y 2 k 5 p14 (x 2 h)2;

(iii) (h, k) 5 (0, 2), p 5 20.5;

(iv) y 2 2 5 212

x2

74. A

75. Sample answer. Rewrite the equation by adding x to each side: y2 1 2y 1 1 5 x 1 5. Then factor the left side:

( y 1 1)2 5 x 1 5 or x 1 5 5 ( y 1 1)2. This

equation is in the form x 2 h 5 p14 ( y 2 k)2 with

p 5 0.25, h 5 25, and k 5 21. So the vertex (h, k) is (25, 21).

LeSSon 29-1 76. a.

Q

P

S R O AB

Q

P

S R O AB

b. C X DY

c. S

RS

P

PS2SR

d. 3 3 PQ

77. a.

BX Y

A

C

BX Y

A

C

b.

D EP

∠C

∠B∠A

78. a.

M N

b. R

S

79. R

S

T

80. C

7© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

LeSSon 29-2

81. H

Q

q m

P

82.

B

C

D

A

r

S

83. a, b, d.

M

b

ac

K

NH

b. See diagram above for construction.

c. line a || line b

d. See diagram above for construction.

e. line c || � ���HK

84. a, b.

T

S

m

n

R

b. See previous diagram for construction.

c. The intersection of m and n is the center of the circle.

d. Check students' work.

85. B

LeSSon 29-3

86.

AB

87. Sample answer.

RP

S

Q

88. a–d.

N

M

P

S

T

QZ

8© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 4 Practice

89.

B

A

P

t

a. See diagram above for construction. Draw AB� ��

. At point B, construct line t so t is perpendicular to AB� ��

.

b. See diagram above for construction. Find point

P on AB� ��

so that BP 5 AB. Then construct circle P with center P and radius PB.

90. A