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Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
5
Beginning-of-Year Diagnostic Test
Use the figure for 1–2.
1. What is the length of ?XY
A 2 3 units C 10 units
B 2 7 units D 14 units
2. What is the midpoint of ?XY
A (1, 1) C (3, 4)
B (2, 2) D ( 2, 2 )
3. In the figure, m 70.KJM
What is the value of x?
A 31 C 41
B 39 D 109
4. Line segment AB with endpoints A(3, 2)
and B(1, 5) is reflected over the x-axis.
What are the coordinates of A' and B'?
A A'(3, 2), B'(1, 5)
B A'(3, 2), B'(1, 5)
C A'(2, 3), B'(5, 1)
D A'(3, 2), B'(1, 5)
5. What is the coordinate notation for the
rule that maps DEF to ?DEF
A (x, y) (x, y)
B (x, y) (x, y)
C (x, y) (y, x)
D (x, y) (y, x)
6. If (x, y) (x 3, y 2), where does
A(4, 6) get mapped to?
A A'(1, 4) C A'(1, 4)
B A'(1, 4) D A'(1, 4)
Use the figure for 7–8.
7. How many lines of symmetry does the
figure have?
A 0 C 2
B 1 D 3
8. What are the angles of rotation less than
360 for the figure?
A 180
B 120, 240
C 90, 180, 270
D 60, 120, 180, 240, 300
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
6
Beginning-of-Year Diagnostic Test
For 9–10, use the figures and the
congruence statement .RST XYZ
9. Which side is congruent to ?RT
A ST C YZ
B XY D XZ
10. What is mY?
A 32 C 65
B 33 D 115
11. What transformation or transformations
can you use to show that quadrilaterals
JKLM and J′K′L′M′ are congruent?
A translation 5 units up
B translation 5 units down
C reflection across the x-axis and
translation 1 unit up
D reflection across the x-axis and
translation 1 unit down
12. In the figure, m2 68. What is m5?
A 22 C 112
B 68 D 122
Use the figure for 13–14.
13. What is the value of x?
A 35 C 105
B 55 D 125
14. What is the value of y?
A 35 C 105
B 55 D 125
15. Which line is parallel to y 3x 4?
A y 3x 1 C 1 43 3
y x
B y 3x 4 D 1
43
y x
16. What is the equation of a line
whose y-intercept is 6 and that is
perpendicular to 2x y 3?
A y 2x 6 C 1
62
y x
B y 2x 6 D 1
62
y x
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
7
Beginning-of-Year Diagnostic Test
17. Which angles are complementary?
A 4 C 1 and 6
B 2 and 3 D 5 and 6
18. In the figure below, .PQR STU
Which statement must be true?
A QR TU C PR TU
B QR SU D QP ST
19. What theorem shows ?DEF JKL
A SSS C ASA
B SAS D AAS
20. What information is needed to show that
ABC XYZ by SAS congruence?
A A X C AB XY
B B Y D AC XZ
21. What information is needed to show that
ABC XYZ by AAS congruence?
A A X C AB XY
B B Y D AC XZ
22. Which congruence statement correctly
shows that the triangles are congruent?
A DEF FGH
B DEF FHG
C DEF HGF
D DEF GHF
23. What is the value of x in the figure?
A 42 C 132
B 48 D 138
24. An equilateral triangle has a perimeter of
12 centimeters. What is the side length of
the triangle?
A 3 cm C 6 cm
B 4 cm D 36 cm
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
8
Beginning-of-Year Diagnostic Test
25. A triangle has side lengths of 7 cm,
10 cm, and 7 cm. Which description best
describes the triangle?
A right C isosceles
B scalene D equilateral
26. Two sides of a triangle measure 8 meters
and 12 meters. What is a possible side
length for the third side?
A 4 m C 20 m
B 10 m D 24 m
27. Which statement is true about the
measures of the angles in ?DEF
A mF mE C mD mE
B mE mF D mD mF
28. A median is a segment that connects the
vertex of a triangle to the midpoint of the
opposite side. What are the coordinates
of the intersection of the three medians of
the triangle?
A (1, 1) C (0, 1)
B (1, 2) D (0, 2)
29. In the figure, MN is the midsegment of
.JKL If KN 7 cm, what is KL?
A 3.5 cm C 14 cm
B 7 cm D 21 cm
30. In the figure, ,LP ,MP and NP are
perpendicular bisectors. Which statement
is not true?
A PH PK C PK PJ
B HM MJ D PL PM
31. In the figure, BD bisects ABC. What is
the value of x?
A 3 C 6
B 5 D 11
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
9
Beginning-of-Year Diagnostic Test
32. Which statement about a parallelogram is
false?
A Diagonals are congruent.
B Diagonals bisect each other.
C Opposite sides are congruent.
D Opposite angles are congruent.
33. A rectangle is shown. If GI 18 cm, what
is GJ?
A 8 cm C 18 cm
B 9 cm D 36 cm
34. PQRS is a parallelogram. What are the
coordinates for the missing vertex S?
A (0, 2) C (1, 2)
B (0, 3) D (4, 2)
35. A parallelogram has vertices
D(1, 1), E(2, 4), F(5, 1), G(2, 2). Which
description best describes the figure?
A rectangle C square
B trapezoid D kite
36. KLMN is a rectangle. What are the
missing coordinates of M?
A (a, a) C (a, b)
B (b, b) D (b, a)
Use the following information for 37–38.
The figure is symmetric about the x-axis.
37. Which description best describes the
figure?
A right triangle
B scalene triangle
C isosceles triangle
D equilateral triangle
38. What is the area of the figure?
A 12 units2 C 36 units2
B 18 units2 D 42 units2
39. What is the perimeter of the figure to the
nearest tenth?
A 12.7 units C 19.4 units
B 18 units D 20.1 units
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
10
Beginning-of-Year Diagnostic Test
40. What is the scale factor of the dilation?
A 14
C 2
B 12
D 4
41. A figure has vertex A(3, 6). The figure is
dilated by a scale factor of 4. The
resulting image is dilated by a scale
factor of 1.
3 What are the coordinates of
vertex A' of the final image?
A (1, 2) C (4, 8)
B (3, 6) D (12, 24)
42. In this figure, .JKL XYZ Which
statement is not true?
A JK LJXY ZX
C LJ KLZX YZ
B KL LJYZ ZX
D KL KJYX YZ
43. Which sequence of transformations maps
ABC to ?RTS
A a translation 4 units right, then a
translation 3 units up
B a translation 3 units right, then a
dilation by a scale factor of 2
C a translation 7 units right, then a
dilation by a scale factor of 2 about S
D a translation 7 units left, then a
dilation by a scale factor of 12
about S
44. In the figure, .CAB EAD What
is AB?
A 9 cm C 15 cm
B 12 cm D 18 cm
45. What is the distance across the river?
A 20 m C 57.6 m
B 30 m D 80 m
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
11
Beginning-of-Year Diagnostic Test
46. Rectangles ABCD and EFGH are similar.
Rectangle ABCD has a length of 6 feet
and a perimeter of 20 feet. Rectangle
EFGH has a length of 18 feet. What is the
perimeter of EFGH ?
A 30 feet C 120 feet
B 60 feet D 216 feet
47. What are the coordinates of point C, such
that 1
?4
AC AB
A (2, 2) C (2, 0)
B (0, 1) D (2, 1)
Use triangle XYZ for 48–49.
48. What is sin X?
A 512
C 1213
B 513
D 1312
49. What is cos X?
A 512
C 1213
B 513
D 1312
50. What is the measure of X to the nearest
tenth of a degree?
A 28.1 C 57.8
B 32.2 D 61.9
51. If BC 4, what is AC?
A 2 C 4 3
B 2 3 D 8
52. What is XZ?
A 11.3 C 27.2
B 21.2 D 51.1
53. What is BC?
A 4.3 C 19.7
B 15.7 D 25.5
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
12
Beginning-of-Year Diagnostic Test
Use the following information for 54–55.
In the circle, m 80AB
54. What is mADB?
A 20 C 80
B 40 D 160
55. What is m ?ADB
A 80 C 160
B 100 D 280
56. In the circle, mACB 98. What is
mAXB?
A 49 C 92
B 82 D 98
57. In the circle, QS and TR intersect at
point P. What is PR?
A 5 C 11
B 9 D 16
Use the circle with center P for 58–59.
58. Find the exact arc length of .QR
A 3
cm C 6 cm
B 3 cm D 18 cm
59. Convert the central angle to radians.
A 3
C 23
B 6
D 32
60. What is the area of the sector formed by
QPR to the nearest tenth of a square
centimeter?
A 4.7 cm2 C 42.4 cm2
B 9.4 cm2 D 84.8 cm2
61. Write an equation of a circle with center
(3, 2) and radius 4.
A 22( 3) 2 16x y
B 22( 3) 2 4x y
C 22( 3) 2 16x y
D 22( 3) 2 4x y
62. Write an equation of a parabola with
focus (0, 2) and directrix y 2.
A 218
y x
B 212
y x
C 22 2y x
D 28 2y x
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
13
Beginning-of-Year Diagnostic Test
63. What is the volume of the prism?
A 120 ft3 C 248 ft3
B 240 ft3 D 480 ft3
64. A rectangular prism has a length of 7 feet
and a width of 5 feet. The volume of the
prism is 280 cubic feet. What is the height
of the prism?
A 8 ft C 40 ft
B 9.3 ft D 56 ft
Use the following information for 65–66.
A cylindrical tank has a diameter of
12 meters and a height of 5 meters.
65. What is the volume of the tank?
A 414.7 m3 C 942.5 m3
B 565.5 m3 D 2261.9 m3
66. What is the surface area of the tank?
A 414.7 m3 C 565.5 m3
B 534.1 m3 D 1281.8 m3
67. What is the volume of the cone?
A 576.0 in3 C 2303.8 in3
B 1073.4 in3 D 6911.5 in3
68. A right circular cone has a slant length of
9 inches and a radius of 4 inches.
What is the surface area of the cone
to the nearest tenth?
A 150.8 in2 C 367.6 in2
B 163.4 in2 D 452.4 in2
69. What is the surface area of a sphere with
a radius of 12 inches?
A 452.4 in3 C 1809.6 in3
B 603.2 in3 D 7238.2 in3
70. A square pyramid with a base area of 16
square feet and a cube with a side length
of 4 feet have equal volumes. What is the
height of the pyramid?
A 4 ft C 32 ft
B 12 ft D 64 ft
71. The trash can shown below is made of
a cylinder and a hemisphere. What is the
volume of the trash can to the nearest
tenth?
A 586.4 in3 C 1172.9 in3
B 1038.8 in3 D 4691.4 in3
72. A solid wooden rectangular prism has
dimensions 3 cm, 2cm, and 5 cm. If it
weighs 22.35 grams, what is the density
of the wood it is made of?
A 0.75 g/cm3 C 1.34 g/cm3
B 0.89 g/cm3 D 2.24 g/cm3
73. A cylinder must have a volume of at least
150 m3 and a radius of 10 m. What is the
minimum possible height of the cylinder?
A 0.2 m C 1.5 m
B 0.5 m D 2.4 m
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
14
Beginning-of-Year Diagnostic Test
74. A paper bag has 10 tiles numbered 1–
10. You randomly choose 3 tiles. What is
the probability that you choose 1, 2, 3 in
that order?
A 310
C 1
720
B 1
120 D
11000
75. There are 12 students on the academic
team. The team advisor randomly
chooses 4 students for the next
competition. What is the probability that
Marcus, Adam, Brittany, and Luisa are
chosen in any order?
A 13
C 1
11,880
B 1
495 D
120,736
76. A jar contains 8 red marbles, 6 yellow
marbles, 4 blue marbles, and 6 green
marbles. One marble is taken from the
jar and not replaced. Then a second
marble is taken from the jar. What is the
probability of choosing a red marble
followed by a blue marble?
A 12
C 132
B 118
D 469
77. A spinner has 4 equal sections
numbered 1–4. A number cube with 6
sides is numbered 1–6. What is the
probability of spinning a 2 and rolling a
4?
A 15
C 512
B 112
D 124
Use the table for 78–79.
Has Pet No Pet Total
Has Sibling 7 7 14
No Sibling 3 5 8
Total 10 12 22
78. Ms. Roberts surveyed the students in her
class about whether they had a sibling or
a pet. The results are shown in the table.
What is the probability that a student has
a sibling if he or she also has a pet?
A 12
C 722
B 710
D 1022
79. What is the probability that a student has
a pet if he or she does not have a
sibling?
A 38
C 318
B 310
D 322
80. A contest allows anyone to enter as
many times as they would like and at the
end of the contest, a winner is randomly
chosen from all the entries. Suppose 450
people enter the contest. If Sabine
entered 3 times and Joe entered 5 times,
then what is the probability Joe or
Sabine will win the contest?
A 0.002 C 0.018
B 0.007 D 0.033
81. A company stores 75% of items in
warehouse A and the rest in warehouse
B. A report found that 1% of items in A
and 2% of items in B are defective. An
item is randomly selected and found to
be defective. What is the probability it
came from A?
A 0.01 C 0.33
B 0.05 D 0.60
Name _______________________________________ Date __________________ Class __________________
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
203
Answer Key
Beginning-of-Year Diagnostic Test
1. C Correct
Substitute the coordinates of the
endpoints into the Distance Formula
to find the length of the line segment.
TEST PREP DOCTOR: Students
who answered A, B, or D should
review the Distance Formula and
operations with integers.
2. A Correct
Substitute the coordinates of the
endpoints into the Midpoint Formula
to find the coordinates of the
midpoint.
TEST PREP DOCTOR: Students
who answered B, C, or D did not use
the Midpoint Formula correctly.
Students should review how the
Midpoint Formula differs from the
Distance Formula.
3. A Correct
In the figure, mKJM mKJL
mLJM. Subtract 39 from 70 to find
the value of x.
TEST PREP DOCTOR: Students
who answered B think the two
smaller angles are congruent.
Students who answered C made a
calculation error. Students who
answered D added the given angle
measurements.
4. B Correct
A reflection across the x-axis maps
a point according to the rule ( , ) ( , ).x y x y
TEST PREP DOCTOR: Students
who answered A, C, or D used an
incorrect rule for a reflection.
Students should review what rules to
use for a reflection depending on the
line of reflection.
5. D Correct
Compare the coordinates of the
vertices of the preimage and the
image. The coordinates of D(1, 2) to
D(2, 1) follows the rule ( , ) ( , ).x y y x
TEST PREP DOCTOR: Students
who answered A, B, or C used an
incorrect rule for a rotation. Students
should review what rules to use for a
rotation depending on angle and
direction of the rotation.
6. B Correct
Based on the rule, this is a translation
three units right and 2 units down.
Add three to the x-coordinate and
subtract 2 from the y-coordinate.
TEST PREP DOCTOR: Students
who answered A, C, or D should
review rules for translations and
operations with integers.
7. D Correct
There is a line of symmetry through
each vertex of the triangle.
TEST PREP DOCTOR: Students
who answered A, B, or C need to
review lines of reflection.
8. B Correct
The angle of rotation is the smallest
angle that maps a figure onto itself.
Identify the fraction of a circle that
the rotation describes, and multiply
it by 360
TEST PREP DOCTOR: Students
who answered A, C, or D need to
review angles of rotation.
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
204
9. D Correct
The congruence statement indicates
that .RT XZ
TEST PREP DOCTOR: Students
who answered A, B, or C should
review congruence statements and
corresponding parts of congruent
triangles.
10. B Correct
The congruence statement indicates
that Y S.
TEST PREP DOCTOR: Students
who answered A, C, or D review
congruence statements and
corresponding parts of congruent
triangles.
11. C Correct
The orientation of the image is
reversed, so there is a reflection
across the x-axis. The figure is then
translated up 1 unit.
TEST PREP DOCTOR: Students
who answered A or B did not
consider that the orientation of the
image is reversed. Students who
answered D reversed the preimage
and the image.
12. C Correct
2 and 4 are corresponding angles,
so they are congruent. 4 and 5
form a linear pair, so they are
supplementary. Therefore, 2 and
5 are supplementary.
TEST PREP DOCTOR: Students
who answered A, B, or D need to
review the angle relationships formed
by parallel lines cut by a transversal.
13. B Correct
The angle that measures x is vertical
to the angle that measures 55, so
the angles are congruent.
TEST PREP DOCTOR: Students
who answered A, C, or D need to
review the angle relationships formed
by intersecting lines and understand
that vertical angles are congruent.
14. D Correct
The angle that measures x forms a
linear pair with the angle that
measures 55, so the angles are
supplementary.
TEST PREP DOCTOR: Students
who answered A, B, or C need to
review the angle relationships formed
by intersecting lines and understand
that the sum of the angles that form a
linear pair is 180.
15. A Correct
The y 3x 4 has a slope of 3.
A line that is parallel to this line will
also have a slope of 3.
TEST PREP DOCTOR: Students
who answered B, C, or D need to
review that parallel lines have the
same slope.
16. C Correct
The 2x y 3 can be written in slope-intercept form as 2 3,y x
which has a slope of 2. A line that is
perpendicular to this line will have a
slope of 12
since 1
2 1.2
TEST PREP DOCTOR: Students
who answered A, B, or D need to
review that the product of the slopes
of perpendicular lines is equal to 1.
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205
17. D Correct
The sum of the measures of
complementary angles is 90.
4, 5, and 6 form a linear pair.
Since 4 is a right angle, the sum of
the measures of 5 and 6 is 90.
TEST PREP DOCTOR: Students
who answered A need to understand
that complementary angles refer to a
pair of angles. Students who answered
B should review the definition of
complementary angles. Students who
answered C should be reminded that
they can only use information that is
given in a problem and should not
rely on how a figure is drawn.
18. A Correct
The congruence statement indicates
that the corresponding sides are
,PQ ST ,QR TU .PR SU
TEST PREP DOCTOR: Students
who answered B, C, or D should
review congruence statements and
corresponding parts of congruent
triangles.
19. B Correct
The triangles have two sides and the
included angle congruent, so the
SAS Theorem can be used to show
that the triangles are congruent.
TEST PREP DOCTOR: Students
who answered A, C, or D should
review the criteria required for
triangle congruence.
20. D Correct
To use SAS, two sides and the
included angle of one triangle need to
be congruent to two sides and the
included angle of another triangle.
C is the included angle for sides
BC and CA and Z is the included
angle for sides YZ and .ZX
TEST PREP DOCTOR: Students
who answered A, B, or C should
review the criteria required for
triangle congruence.
21. A Correct
To use AAS, two angles and the non-
included angle of one triangle need to
be congruent to two angles and the
non-included angle of another
triangle. BC is the non-included side
for A and C and YZ is the non-
included side for X and Z.
TEST PREP DOCTOR: Students
who answered B should review the
difference between an included side
and a non-included side. Students
who answered C or D should review
the criteria required for triangle
congruence.
22. D Correct
In the figure, F F, so the
triangles are congruent by SAS. Use
the corresponding parts to write the
congruence statement.
TEST PREP DOCTOR: Students
who answered A, B, or C should
review congruence statements and
corresponding parts of congruent
triangles.
23. D Correct
The angle that measures x is an
exterior angle. The measure of an
exterior angle is equal to the sum of
its remote interior angles. The remote
interior angles measure 90 and 48.
TEST PREP DOCTOR: Students
who answered A used the Triangle
Sum Theorem to find the missing
interior angle. Students who
answered B determined that x was
congruent to the given angle, 48.
Students who answered C
determined that x was supplementary
to the given angle, 48. Students
should review the relationships
between the interior and exterior
angles of a triangle.
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206
24. B Correct
An equilateral triangle has three
congruent sides. To find the side
length, divide the perimeter by 3.
TEST PREP DOCTOR: Students
who answered A or C may have
made a calculation error. Students
who answered D multiplied the
perimeter by 3 instead of dividing
by 3. Students should review the
properties of equilateral triangles.
25. C Correct
The triangle has two congruent sides,
so it is isosceles.
TEST PREP DOCTOR: Students
who answered A, B, or D should
review how to classify a triangle by
its sides.
26. B Correct
The Triangle Inequality Theorem
states that any two sides of a triangle
are greater than the third side.
Therefore, the length of the third side
of the triangle must be greater than
12 8 4 and less than 12 8 20.
TEST PREP DOCTOR: Students
who answered A or C need to be
reminded that two sides of a triangle
must be greater than, not greater
than or equal to, the third side.
Students who answered D should
review the Triangle Inequality
Theorem.
27. A Correct
In a triangle, the larger angle is
opposite the longer side.
TEST PREP DOCTOR: Students
who answered B, C, or D should
review side-angle relationships in
triangles.
28. A Correct
Draw each median of the triangle and
find the point of intersection. The
medians lie on the lines x 1,
y 1, and y 2x 2.
TEST PREP DOCTOR: Students
who answered B, C, or D should
review how to find the medians of
a triangle.
29. C Correct
The midsegment connects the
midpoints of two sides of the triangle,
so KN KL.
TEST PREP DOCTOR: Students
who answered A may have used
KL 7, and found KN. Students who
answered B found NL, but did not
follow through with finding the KL.
Students who answered D need
to review the properties of a
midsegment of a triangle.
30. D Correct
Each perpendicular bisector
intersects a side of the triangle at its
midpoint. Point P is the intersection
of the three perpendicular bisectors,
so it is equidistant to H, J, and K.
TEST PREP DOCTOR: Students
who answered A need to review the
definition of a perpendicular bisector
and understand that it intersects a
line segment at its midpoint. Students
who answered B or C need to
understand that that the point of
concurrency of the three
perpendicular bisectors of a triangle
is equidistant from the vertices of the
triangle.
31. B Correct
Since point D is on the bisector of the
angle, it is equidistant from the sides
of the angle. So, AD DC. To find
the value of x, solve the equation
2x 6 16.
TEST PREP DOCTOR: Students
who answered D solved the equation
incorrectly. Students who answered
A or C need to review the properties
of angle bisectors.
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
207
32. A Correct
The diagonals of a parallelogram are
not congruent unless it is a rectangle.
TEST PREP DOCTOR: Students
who answered B, C, or D should
review properties of parallelograms.
33. B Correct
The diagonals of a rectangle bisect
each other and are congruent.
TEST PREP DOCTOR: Students
who answered C or D may have
misread the given information.
Students who answered D should
review properties of rectangles and
understand that the diagonals bisect
each other.
34. A Correct
PS must be parallel and congruent
to .QR
TEST PREP DOCTOR: Students
who answered B may have misread
the graph since the height of the
parallelogram is 3 units. Students
who answered C or D should review
properties of parallelograms and
understand how to determine the
length and slope of a line segment on
the coordinate plane using the
Distance Formula and Slope
Formula.
35. C Correct
One way to solve the problem is to
graph the vertices. The diagonals of
the parallelogram are perpendicular
and congruent, so the parallelogram
is a square.
TEST PREP DOCTOR: Students
who answered A did not choose the
best description for the figure.
Students who answered B or D
should review properties of
quadrilaterals and understand how to
determine the length and slope of a
line segment on the coordinate plane
using the Distance Formula and
Slope Formula.
36. C Correct
Opposite sides of a rectangle are
congruent and all four angles are
right angles. The rectangle has
length a and width b, so the
x-coordinate is a and the
y-coordinate is b.
TEST PREP DOCTOR: Students
who answered D reversed the x-
and y-coordinates. Students who
answered A or B should review how
to position a figure on the coordinate
plane and assign coordinates to each
vertex.
37. C Correct
Use the Distance Formula to find
each length: DE 45, EF 45, and FD 6. Since the triangle has
two congruent sides, the triangle is
isosceles.
TEST PREP DOCTOR: Students
who answered A, B, or D need to
review how to use the Distance
Formula to find the length of a line
segment on the coordinate plane.
Students may also need to review
how to classify a triangle by its sides.
38. B Correct
Use the formula A 12
bh. The base
of the triangle is 6 units and the
height is 6 units.
TEST PREP DOCTOR: Students
who answered A added the length of
the base and height of the triangle,
and students who answered B did
not multiply the product of the base
and height by 1.
2 These students
should review the formula for the
area of a triangle. Students who
answered D need to review how to
use the Distance Formula to find the
length of a line segment on the
coordinate plane.
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208
39. C Correct
Use the Distance Formula to find
each length: DE 45, EF 45, and FD 6. Add the sides to find the
perimeter.
TEST PREP DOCTOR: Students
who answered A only added 2 sides
of the triangle. Students who
answered B used a side length of 6
units for each side and students who
answered D used a side length of
45 for each side. Review how to
use the Distance Formula to find the
length of a line segment on the
coordinate plane.
40. B Correct
Compare the coordinates of the
image to the preimage. The dilation
follows the rule 1 1
( , ) , .2 2
x y x y
TEST PREP DOCTOR: Students
who answered A may have misread
the coordinates of the vertices of the
triangles. Students who answered C
made an error in determining which
figure is the preimage and which
figure is the image. Students who
answered D need to review how to
find a scale factor of a dilation on the
coordinate plane.
41. C Correct
Perform the dilation ( , ) (4 , 4 )x y x y
followed by the dilation The dilation
follows the rule 1 1
( , ) , .3 3
x y x y
Both dilations can be combined into a
single transformation. The dilation
follows the rule 4 4
( , ) , .3 3
x y x y
TEST PREP DOCTOR: Students
who answered A or D only performed
one of the dilations. Students who
answered B need to review how to
perform dilations using coordinates.
42. D Correct
The similarity statement and angle arcs indicate that ,J X
,K Y L Z and,
.JK KL LJXY YZ ZX
TEST PREP DOCTOR: Students
who answered A, B, or C need to
review how to set up proportions
using the corresponding sides of
similar triangles.
43. C Correct
First map point C to point S by a
translation 7 units right. Then dilate
the figure by a scale factor of 2.
TEST PREP DOCTOR: Students who answered A may have confused the side lengths of the triangles with the change in coordinates for a translation. Students who answered B did not translate the vertices of the preimage to the corresponding vertices of the image. Students who answered D mapped RTS to ABC.
44. C Correct
Since ~ ,CAB EAD .ED ADCB AB
Solve the proportion 4 6
.10 AB
TEST PREP DOCTOR: Students
who answered A found the length of
.DB Students who answered B or D
should review how to set up a
proportion for corresponding sides of
similar figures, and understand that a
line parallel to one side of a triangle
that intersects the other two sides,
divides the triangle proportionally.
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
209
45. A Correct
Use similar triangles to set up and
solve the proportion 48 24
.40 JK
TEST PREP DOCTOR: Students
who answered B, C, or D did not
write a correct proportion for the
diagram. Review how to solve
indirect measurement problems using
similar triangles.
46. B Correct
Use the given length and perimeter to
find the width of rectangle ABCD.
The dimensions of rectangle ABCD
are 6 by 4. Use corresponding parts
of similar figures to find the width of
rectangle EFGH. The dimensions of
rectangle EFGH are 18 by 12. Use
these dimensions to calculate the
perimeter of EFGH.
TEST PREP DOCTOR: Students
who answered A found the correct
length and width of EFGH, but only
added the two sides to find the
perimeter. Students who answered C
found the width of ABCD to be
20 6 14 and did not divide by 2 to
find the correct length and width of
ABCD. Students who answered D
found the area of EFGH. Review
how to make a plan to solve word
problems with more than one step.
47. A Correct
Look at the rise and run between the endpoints of the line segment. The rise is
4 units and the run is 8 units. Find 14
of
each distance: 1•4 1
4 (rise) and
1•8 2
4 (run). So, the coordinates of
point C will be 1 unit up and 2 units right from point A.
TEST PREP DOCTOR: Students who answered B or C found the coordinates
of point C to be 12
AB and 34
AB,
respectively. Students who answered D did not find a point on the line. Review how to find the coordinates of a point that is a fractional distance from one end of a line segment.
48. B Correct
Use the sine ratio, sin .YZ
XXY
TEST PREP DOCTOR: Students who answered A, C, or D should review the definitions of trigonometric ratios.
49. C Correct
Use the cosine ratio, cos .XZ
XXY
TEST PREP DOCTOR: Students who answered A, B, or D should review the definitions of trigonometric ratios.
50. A Correct
Since 8
tan ,15
X use the inverse
tangent ratio, mX 1 8tan .
15
TEST PREP DOCTOR: Students who answered B, C, or D should review the definitions of trigonometric ratios and inverse trigonometric ratios.
51. C Correct
In a 30-60-90 triangle, the hypotenuse is twice as long as the
shorter leg and the longer leg is 3
times as long as the shorter leg.
TEST PREP DOCTOR: Students who answered A, B, or D should review the
properties of 30-60-90 right triangles.
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
210
52. B Correct
Since XZ is opposite Y and XY is
the hypotenuse of the right triangle,
you can solve the equation
sin6224x
, where x represents the
length of .XZ
TEST PREP DOCTOR: Students
who answered A or D used cos 62
in their calculations. Students who
answered C did not set up the
equation correctly. Review how to
use trigonometric ratios and inverse
trigonometric ratios to solve right
triangles.
53. A Correct
Using the Law of Sines,
sin(78 ) sin(50 )20 BC
. Cross multiplying,
20sin(50 )15.7.
sin(78 )BC
TEST PREP DOCTOR: Students who answered A attempted to use right angle trigonometry. Students who answered D used the Law of Sines but did not math angles with opposite sides. Students who answered C should review solving triangles in general.
54. B Correct
The measure of an inscribed angle is
equal to half the measure of its
intercepted arc. So,
1m m .
2ADB AB
TEST PREP DOCTOR: Students
who answered D may have reversed
the measures of the inscribed angle
and the arc. Students who answered
C may have confused the properties
of inscribed angles with the
properties of central angles. Students
who answered A should review the
properties of inscribed angles.
55. D Correct
In the figure, ADB is a major arc and
AB is a minor arc. So,
m 360 .ADB AB
TEST PREP DOCTOR: Students
who answered A, B, or C should
review the properties of minor arcs,
major arcs, and arc addition.
56. B Correct
A circumscribed angle of a circle and
its associated central angle are
supplementary. So,
m m 180.AXB ACB
TEST PREP DOCTOR: Students
who answered A, C, or D should
review the properties of tangents,
circumscribed angles, and central
angles.
57. B Correct
If two chords intersect inside a circle,
then the products of the lengths of
the segments of the chords are
equal. So, QP PS TP PR.
TEST PREP DOCTOR: Students
who answered A, C, or D should
review how segments in circles are
related.
58. B Correct
The arc length, s, of an arc with
measure m and radius r is given by
the formula 2 .360m
s r
TEST PREP DOCTOR: Students who
answered C used the diameter in their
calculations. Students who answered D
found the circumference of the circle.
Students who answered A should
review how to find arc length using the
measure of a central angle and the
radius of a circle. Students should also
understand the proportional relationship
between arc length and the
circumference of a circle.
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
211
59. A Correct
To convert to a radian measure, let m
represent the degree measure of an
angle and evaluate the expression
•2 .360m
TEST PREP DOCTOR: Students
who answered B, C, or D reduced
60360
incorrectly or did not multiply by
2. Students should review how to
convert degrees to radians and
radians to degrees.
60. C Correct
The area A of a sector with a central
angle of m of a circle with radius r is
2.360m
A r
TEST PREP DOCTOR: Students
who answered A did not square the
radius. Students who answered D
reduced 60360
incorrectly. Students
who answered D should review how
to find the area of a sector using the
measure of a central angle and the
radius of a circle. Students should
also understand the proportional
relationship between the area of a
sector of a circle and the area of the
circle.
61. C Correct
The equation of a circle with center
(h, k) and radius r is 2 2 2( ) ( ) .x h y k r
TEST PREP DOCTOR: Students
who answered A did not subtract the
coordinates of the center of the circle.
Students who answered D did not
square the radius. Students who
answered B did not subtract the
coordinates and did not square the
radius. Review how to write an
equation of a circle using the
coordinates of the center and the
length of the radius.
62. A Correct
The general form of a parabola of this
type is 214
y xp
and p 2.
Therefore 2 21 1.
4(2) 8y x x
TEST PREP DOCTOR: Students
who answered B, C, or D should
review how to find the equation of
a parabola given the focus and
directrix.
63. B Correct
The formula for the volume of a prism
is V Bh. The base is a triangle, so
1•6•8
2B and h 10.
TEST PREP DOCTOR: Students
who answered A or D did not use the
correct formula to find the area of the
triangular base. Students who
answered C need to review the
formula for the volume of a triangular
prism.
64. A Correct
The formula for the volume of a prism
is V Bh. The volume and base area
are given, so divide the volume by
the base area to find the height of the
prism.
TEST PREP DOCTOR: Students
who answered B made a calculation
error in finding the base area.
Students who answered C or D
did not divide the volume by both
dimensions of the base. Review
the formula for the volume of a
rectangular prism and how to
solve for a variable.
65. B Correct
The formula for the volume of a
cylinder is V r2h. Substitute r 6
and h 5 into the formula to find the
volume of the tank.
TEST PREP DOCTOR: Students
who answered A found the surface
area. Students who answered C
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212
switched the values for the diameter
and the height. Students who
answered D used the value for the
diameter in their calculations. Review
the formula for the volume of a
cylinder and remind students to find
the value of the radius when they are
given the diameter.
66. A Correct
The formula for the surface area of a
cylinder is S 2rh 2r 2. Substitute
r 6 and h 5 into the formula to find
the surface area of the tank.
TEST PREP DOCTOR: Students
who answered B switched the values
for the diameter and the height.
Students who answered D used the
value for the diameter in their
calculations. Students who answered
C should review the formula for the
volume of a cylinder.
67. C Correct
The formula for the volume of a cone
is V 13r 2h. Substitute r 10 and
h 22 into the formula.
TEST PREP DOCTOR: Students
who answered D did not multiply by
13
. Students who answered A or B
should review the formula for the
volume of a cone.
68. B Correct
The formula for the surface area of a
cone is S r r 2. Substitute
r 4 and 9 into the formula.
TEST PREP DOCTOR: Students
who answered A used the given
values in the formula for the volume
of a cone. Students who answered C
switched the values for the radius
and slant height. Students who
answered D should review the
formula for the surface area of
a cone.
69. C Correct
The formula for the surface area of a
sphere is S 4 r 2. Substitute r 12
into the formula.
TEST PREP DOCTOR: Students
who answered A used 12
r in their
calculations. Students who answered
B found the 13
surface area of a
sphere. Students who answered D
found the volume of a sphere.
Review the formula for the surface
area of a sphere.
70. B Correct
One way to solve this problem is to
recognize that the pyramid and the
cube have the same base area.
Since a pyramid is 13
the volume of a
cube with the same base area, the
height of the pyramid must be 3 times
the height of the cube.
TEST PREP DOCTOR: Students
who answered A did not consider
that the volume of a pyramid is 13
times the product of the base area
and the height. Students who
answered C or D should review the
formulas for the volume of a pyramid
and the volume of a cube and
understand the relationship between
the volumes of these figures.
71. B Correct
Use 2V r h for the volume of a
cylinder and V 31 42 3
r
for the
volume of the hemisphere. Substitute
r 4 and h 18 into the appropriate
formulas.
TEST PREP DOCTOR: Students
who answered C used the volume of
a sphere instead of a hemisphere.
Students who answered D used the
value for the diameter in their
calculations. Students who answered
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213
A should review the formulas for
cylinders, spheres, hemispheres, and
how to find the volume of composite
figures.
72. A Correct
The volume of the prism is 3 2 5
30 cm3. Using this and the given
mass in the density formula
massD
volume results in the value 0.75.
TEST PREP DOCTOR: Students
who answered B or D should review
the formula for volume of a prism.
Students who answered C should
review the formula for the density.
73. B Correct
The minimum height will occur when
the volume is at its minimum.
Therefore, 2150 10 h . Solving
this formula for h results in h 0.5.
TEST PREP DOCTOR: Students
who answered A used the correct
volume formula but did not replace
values correctly. Students who answered C did not include in their
calculations. Students who answered
D should review the volume formula
for a cylinder.
74. C Correct
Since order is important, the number
of possible outcomes is the
permutation of 10 objects taken 3 at
a time. So, the probability is 10 3
1.
P
TEST PREP DOCTOR: Students
who answered A found the simple
probability of choosing 1, 2, or 3 out
of 10 tiles. Students who answered B
used a combination instead of a
permutation. Students who answered
D found the probability of
independent events. Students should
review permutations and
combinations.
75. B Correct
Since order is not important, the
number of possible outcomes is the
combination of 12 objects taken 4 at
a time. So, the probability is 12 4
1.
C
TEST PREP DOCTOR: Students
who answered A found the simple
probability of choosing any 4 out of
12 students. Students who answered
C found the probability of dependent
events. Students who answered D
used a permutation instead of a
combination. Students should review
permutations and combinations.
76. D Correct
These are dependent events, so
and ( ) ( | )P A B P A PB A . There
are 24 marbles in the bag. Since the
first marble is not replaced, there are
23 marbles in the bag for the second
draw. The probability is 8 4
.24 23
TEST PREP DOCTOR: Students
who answered B found the
probability with replacement.
Students who answered A or C
should review probability of
dependent events.
77. D Correct
These are independent events, so
and P A B P A P B . The
probability of spinning a 2 is 14
and
the probability of rolling a 4 is 1.
6 So,
the probability of both events is 1 14 6
.
TEST PREP DOCTOR: Students
who answered C added the
probabilities of each event. Students
who answered A or B should review
probability of independent events.
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
214
78. B Correct
The formula for conditional probability
is ( )
( | ) .( )
P A BPB A
P A
Look at the
column for students who have a pet.
There are 10 students. Of those 10
students, 7 students also have a
sibling.
TEST PREP DOCTOR: Students
who answered A found the
probability that a student has a pet
given that the student has a sibling.
Students who answered C found the
probability of students who have both
a pet and a sibling. Students who
answered D found the probability that
a student has a pet. Students should
review conditional probability.
79. A Correct
The formula for conditional probability
is ( )
( | )( )
P A BPB A
P A
. Look at the
row for students who do not have
a sibling. There are 8 students. Of
those 8 students, 3 students have
a pet.
TEST PREP DOCTOR: Students
who answered B found the
probability that a student does not
have a sibling given that the student
has a pet. Students who answered C
or D should review conditional
probability.
80. C Correct
The two possibilities are disjoint so ( ) ( ) ( )P A B P A PB . The
probability Sabine wins is 3
450 and
the probability Joe wins is 5
.450
TEST PREP DOCTOR: Students
who answered A found the
probability that a single entry would
win. Students who answered B found
the probability that Sabine would win.
Students who answered D multiplied
the probabilities for Sabine and Joe.
81. D Correct
Using Bayes Formula,
0.75(0.01)( | )
0.75(0.01) 0.25(0.02)0.60
P A def
.
TEST PREP DOCTOR: Students
who answered A found the
probability that a randomly selected
item is defective. Students who
answered B used the percentages for
the opposite warehouses. Students
who answered C used 0.75 instead
of 0.25 in the denominator.