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