Geometry Multiple Choice Regents Exam Questions

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Geometry Multiple Choice Regents Exam Questions

1 What are the center and the radius of the circle 5 In the diagram below, lines n and m are cut by
whose equation is (x − 5) 2 + (y + 3) 2 = 16? transversals p and q.
1) (−5, 3) and 16
2) (5, −3) and 16
3) (−5, 3) and 4
4) (5, −3) and 4

2 The number of degrees in the sum of the interior

angles of a pentagon is
1) 72 What value of x would make lines n and m parallel?
2) 360 1) 110
3) 540 2) 80
4) 720 3) 70
4) 50

3 The coordinates of the endpoints of AB are A(0, 0)

and B(0, 6) . The equation of the perpendicular 6 In the diagram below, LATE is an isosceles
bisector of AB is trapezoid with LE ≅ AT , LA = 24 , ET = 40 , and
1) x = 0 AT = 10 . Altitudes LF and AG are drawn.
2) x = 3
3) y = 0
4) y = 3

4 In scalene triangle ABC, m∠B = 45 and m∠C = 55.

What is the order of the sides in length, from What is the length of LF ?
longest to shortest? 1) 6
1) AB , BC , AC 2) 8
3) 3
2) BC , AC , AB 4) 4
3) AC , BC , AB
4) BC , AB , AC

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7 In the diagram below of ABC , BC is extended to 10 In the diagram below of ABC , AB ≅ AC ,

D. m∠A = 3x, and m∠B = x + 20.

What is the value of x?

1) 10
2) 28
If m∠A = x 2 − 6x, m∠B = 2x − 3, and 3) 32
m∠ACD = 9x + 27, what is the value of x? 4) 40
1) 10
2) 2
3) 3
4) 15
11 Quadrilateral MNOP is a trapezoid with MN  OP.
If M ′N ′O ′P ′ is the image of MNOP after a
reflection over the x-axis, which two sides of
quadrilateral M ′N ′O ′P ′ are parallel?
8 In rhombus ABCD, the diagonals AC and BD
intersect at E. If AE = 5 and BE = 12 , what is the 1) M ′N ′ and O ′P ′
length of AB ? 2) M ′N ′ and N ′O ′
1) 7 3) P ′M ′ and O ′P ′
2) 10 4) P ′M ′ and N ′O ′
3) 13
4) 17

12 A sphere is inscribed inside a cube with edges of 6

cm. In cubic centimeters, what is the volume of the
9 Scalene triangle ABC is similar to triangle DEF. sphere, in terms of π ?
Which statement is false? 1) 12π
1) AB :BC = DE :EF 2) 36π
2) AC :DF = BC :EF 3) 48π
3) ∠ACB ≅ ∠DFE 4) 288π
4) ∠ABC ≅ ∠EDF

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Geometry Multiple Choice Regents Exam Questions
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13 Which line is parallel to the line whose equation is 16 As shown on the graph below, R ′S ′T ′ is the
4x + 3y = 7 and also passes through the point image of RST under a single transformation.
(−5, 2)?
1) 4x + 3y = −26
2) 4x + 3y = −14
3) 3x + 4y = −7
4) 3x + 4y = 14

14 In a given triangle, the point of intersection of the

three medians is the same as the point of
intersection of the three altitudes. Which
classification of the triangle is correct?
1) scalene triangle
2) isosceles triangle
3) equilateral triangle Which transformation does this graph represent?
4) right isosceles triangle 1) glide reflection
2) line reflection
3) rotation
4) translation
15 Line n intersects lines l and m, forming the angles
shown in the diagram below.

17 In the diagram below of circle O, PA is tangent to

circle O at A, and PBC is a secant with points B
and C on the circle.

Which value of x would prove l  m?

1) 2.5
2) 4.5
3) 6.25
4) 8.75 If PA = 8 and PB = 4 , what is the length of BC ?
1) 20
2) 16
3) 15
4) 12

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Geometry Multiple Choice Regents Exam Questions
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18 Which type of triangle can be drawn using the 21 What is an equation of circle O shown in the graph
points (−2, 3), (−2, −7), and (4, −5)? below?
1) scalene
2) isosceles
3) equilateral
4) no triangle can be drawn

19 In the diagram below of rhombus ABCD,

m∠C = 100.

1) (x + 1) 2 + (y − 3) 2 = 25
2) (x − 1) 2 + (y + 3) 2 = 25
3) (x − 5) 2 + (y + 6) 2 = 25
4) (x + 5) 2 + (y − 6) 2 = 25

What is m∠DBC ?
1) 40
2) 45 22 In the diagram below, AB , BC , and AC are
3) 50 tangents to circle O at points F, E, and D,
4) 80 respectively, AF = 6 , CD = 5, and BE = 4 .

20 In the diagram below of ABCD , AC ≅ BD .

Using this information, it could be proven that

1) BC = AB
2) AB = CD What is the perimeter of ABC ?
3) AD − BC = CD 1) 15
4) AB + CD = AD 2) 25
3) 30
4) 60

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23 Which reason could be used to prove that a 26 The angle formed by the radius of a circle and a
parallelogram is a rhombus? tangent to that circle has a measure of
1) Diagonals are congruent. 1) 45°
2) Opposite sides are parallel. 2) 90°
3) Diagonals are perpendicular. 3) 135°
4) Opposite angles are congruent. 4) 180°

24 What is the equation of a line passing through 27 In FGH , m∠F = 42 and an exterior angle at
(2, −1) and parallel to the line represented by the vertex H has a measure of 104. What is m∠G?
equation y = 2x + 1? 1) 34
2) 62
1) y = − 1 x 3) 76
2
4) 146
2) y = − 1 x + 1
2
3) y = 2x − 5
4) y = 2x − 1
28 In the diagram below of right triangle ABC, altitude
BD is drawn to hypotenuse AC , AC = 16 , and
CD = 7.
25 In the diagram below, ABC is circumscribed
about circle O and the sides of ABC are tangent
to the circle at points D, E, and F.

What is the length of BD ?

1) 3 7
If AB = 20 , AE = 12 , and CF = 15, what is the 2) 4 7
length of AC ? 3) 7 3
1) 8 4) 12
2) 15
3) 23
4) 27

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Geometry Multiple Choice Regents Exam Questions
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29 Pentagon PQRST has PQ parallel to TS . After a 32 In the diagram below, A ′B ′C ′ is a transformation

translation of T 2, −5 , which line segment is parallel of ABC , and A ″B ″C ″ is a transformation of
A ′B ′C ′.
to P ′Q ′?
1) R ′Q ′
2) R ′S ′
3) T ′S ′
4) T ′P ′

30 The slope of line  is − 1 . What is an equation of a

3
line that is perpendicular to line ?
1) y + 2 = 1 x
3
2) −2x + 6 = 6y
3) 9x − 3y = 27 The composite transformation of ABC to
4) 3x + y = 0 A ″B ″C ″ is an example of a
1) reflection followed by a rotation
2) reflection followed by a translation
3) translation followed by a rotation
31 The vertices of the triangle in the diagram below 4) translation followed by a reflection
are A(7, 9) , B(3, 3) , and C(11, 3).

33 Which equation represents the line parallel to the

line whose equation is 4x + 2y = 14 and passing
through the point (2, 2)?
1) y = −2x
2) y = −2x + 6
3) y = 1 x
2
4) y = 1 x + 1
2

What are the coordinates of the centroid of ABC ?

1) (5, 6)
2) (7, 3)
3) (7, 5)
4) (9, 6)

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Geometry Multiple Choice Regents Exam Questions
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34 In the diagram below of circle O, chord AB is 36 A man wants to place a new bird bath in his yard so
that it is 30 feet from a fence, f, and also 10 feet
parallel to chord CD.
from a light pole, P. As shown in the diagram
below, the light pole is 35 feet away from the
fence.

Which statement must be true?

1) AC ≅ BD How many locations are possible for the bird bath?
2) AB ≅ CD 1) 1
3) AB ≅ CD 2) 2
3) 3
4) ABD ≅ CDB 4) 0

35 What is an equation of circle O shown in the graph 37 In the diagram below, LMO is isosceles with
below? LO = MO .

If m∠L = 55 and m∠NOM = 28, what is m∠N ?

1) 27
2) 28
3) 42
1) (x + 2) 2 + (y − 2) 2 = 9 4) 70
2) (x + 2) 2 + (y − 2) 2 = 3
3) (x − 2) 2 + (y + 2) 2 = 9
4) (x − 2) 2 + (y + 2) 2 = 3

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Geometry Multiple Choice Regents Exam Questions
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38 When solved graphically, what is the solution to 41 The vertices of parallelogram ABCD are A(2, 0) ,
the following system of equations? B(0, −3) , C(3, −3), and D(5, 0) . If ABCD is
y = x 2 − 4x + 6 reflected over the x-axis, how many vertices remain
y=x+2
invariant?
1) 1
2) 2
1) (1, 4)
3) 3
2) (4, 6)
4) 0
3) (1, 3) and (4, 6)
4) (3, 1) and (6, 4)

42 Which quadrilateral has diagonals that always

bisect its angles and also bisect each other?
39 For a triangle, which two points of concurrence 1) rhombus
could be located outside the triangle? 2) rectangle
1) incenter and centroid 3) parallelogram
2) centroid and orthocenter 4) isosceles trapezoid
3) incenter and circumcenter
4) circumcenter and orthocenter

43 In the diagram below of ABC , D is the midpoint

of AB , and E is the midpoint of BC .
40 In AED with ABCD shown in the diagram below,
EB and EC are drawn.

If AB ≅ CD, which statement could always be

proven?
1) AC ≅ DB
2) AE ≅ ED
3) AB ≅ BC If AC = 4x + 10 , which expression represents DE?
4) EC ≅ EA 1) x + 2.5
2) 2x + 5
3) 2x + 10
4) 8x + 20

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44 What is the equation of a circle whose center is 4 48 In the diagram below of circle O, chord AB is
units above the origin in the coordinate plane and
parallel to chord GH . Chord CD intersects AB at E
whose radius is 6?
and GH at F.
1) x 2 + (y − 6) 2 = 16
2) (x − 6) 2 + y 2 = 16
3) x 2 + (y − 4) 2 = 36
4) (x − 4) 2 + y 2 = 36

45 Segment AB is the diameter of circle M. The

coordinates of A are (−4, 3). The coordinates of M
are (1, 5). What are the coordinates of B?
1) (6, 7)
2) (5, 8) Which statement must always be true?
3) (−3, 8) 1) AC ≅ CB
4) (−5, 2) 2) DH ≅ BH
3) AB ≅ GH
4) AG ≅ BH

46 In DEF , m∠D = 3x + 5, m∠E = 4x − 15, and

m∠F = 2x + 10. Which statement is true?
1) DF = FE
49 Given three distinct quadrilaterals, a square, a
2) DE = FE
rectangle, and a rhombus, which quadrilaterals
3) m∠E = m∠F
must have perpendicular diagonals?
4) m∠D = m∠F
1) the rhombus, only
2) the rectangle and the square
3) the rhombus and the square
4) the rectangle, the rhombus, and the square
←→ ←→
47 If AB is contained in plane P, and AB is
perpendicular to plane R, which statement is true?
←→
1) AB is parallel to plane R. 50 The diameter of a sphere is 15 inches. What is the
2) Plane P is parallel to plane R. volume of the sphere, to the nearest tenth of a
←→ cubic inch?
3) AB is perpendicular to plane P. 1) 706.9
4) Plane P is perpendicular to plane R. 2) 1767.1
3) 2827.4
4) 14,137.2

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51 What is the slope of a line perpendicular to the line 54 As shown in the diagram below, ABC ∼ DEF ,
whose equation is 20x − 2y = 6? AB = 7x , BC = 4 , DE = 7 , and EF = x .
1) −10
2) − 1
10
3) 10
1
4)
10

52 What is the slope of a line that is perpendicular to

the line whose equation is 3x + 5y = 4?
1) − 3 What is the length of AB ?
5
3
1) 28
2) 2) 2
5
3) 14
3) − 5 4) 4
3
5
4)
3

55 A line segment has endpoints A(7, −1) and B(−3, 3) .

What are the coordinates of the midpoint of AB ?
53 In the diagram below, EF is the median of 1) (1, 2)
trapezoid ABCD. 2)  2, 1 
3) (−5, 2)
4)  5, −2 

56 If JKL ≅ MNO, which statement is always true?

1) ∠KLJ ≅ ∠NMO
2) ∠KJL ≅ ∠MON
If AB = 5x − 9 , DC = x + 3 , and EF = 2x + 2 , what
is the value of x? 3) JL ≅ MO
1) 5 4) JK ≅ ON
2) 2
3) 7
4) 8

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57 In the diagram below of circle O, diameter AB is 59 When writing a geometric proof, which angle
relationship could be used alone to justify that two
parallel to chord CD.
angles are congruent?
1) supplementary angles
2) linear pair of angles
3) adjacent angles
4) vertical angles

60 The coordinates of point A are (−3a, 4b). If point A'

is the image of point A reflected over the line y = x,
the coordinates of A' are
If mCD = 70, what is mAC ? 1) (4b, −3a)
1) 110 2) (3a, 4b)
2) 70 3) (−3a, −4b)
3) 55 4) (−4b, −3a)
4) 35

61 In the diagram below, ABC ≅ XYZ .

58 In the diagram below of ABC , side BC is
extended to point D, m∠A = x, m∠B = 2x + 15, and
m∠ACD = 5x + 5.

Which statement must be true?

1) ∠C ≅ ∠Y
2) ∠A ≅ ∠X
3) AC ≅ YZ
What is m∠B?
1) 5 4) CB ≅ XZ
2) 20
3) 25
4) 55

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62 What is an equation of the circle with a radius of 5 65 What is the measure of each interior angle of a
and center at (1, −4)? regular hexagon?
1) (x + 1) 2 + (y − 4) 2 = 5 1) 60°
2) 120°
2) (x − 1) 2 + (y + 4) 2 = 5 3) 135°
3) (x + 1) 2 + (y − 4) 2 = 25 4) 270°
4) (x − 1) 2 + (y + 4) 2 = 25

66 The volume of a rectangular prism is 144 cubic

inches. The height of the prism is 8 inches. Which
63 What is an equation of the line that passes through
measurements, in inches, could be the dimensions
the point (−2, 3) and is parallel to the line whose
of the base?
equation is y = 3 x − 4? 1) 3.3 by 5.5
2
2) 2.5 by 7.2
−2
1) y = x 3) 12 by 8
3
4) 9 by 9
2) y = −2 x + 5
3 3
3
3) y = x
2
←→
4) y = 3 x + 6 67 In the diagram below, AB is perpendicular to plane
2
AEFG.

64 In the diagram below of BCD, side DB is

extended to point A.

Which statement must be true?

1) m∠C > m∠D Which plane must be perpendicular to plane
2) m∠ABC < m∠D AEFG?
3) m∠ABC > m∠C 1) ABCE
4) m∠ABC > m∠C + m∠D 2) BCDH
3) CDFE
4) HDFG

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68 In parallelogram ABCD shown below, diagonals 70 What is the slope of a line that is perpendicular to
AC and BD intersect at E. the line represented by the equation x + 2y = 3?
1) −2
2) 2
3) − 1
2
4) 1
2

Which statement must be true?

1) AC ≅ DB 71 Plane A is parallel to plane B. Plane C intersects
2) ∠ABD ≅ ∠CBD plane A in line m and intersects plane B in line n.
3) AED ≅ CEB Lines m and n are
4) DCE ≅ BCE 1) intersecting
2) parallel
3) perpendicular
4) skew
69 The graph below shows JT and its image, J ′T ′,
after a transformation.

72 Which equation represents the perpendicular

bisector of AB whose endpoints are A(8, 2) and
B(0, 6) ?
1) y = 2x − 4
2) y = − 1 x + 2
2
3) y = − 1 x + 6
2
4) y = 2x − 12

73 An equation of the line that passes through (2, −1)

Which transformation would map JT onto J ′T ′?
and is parallel to the line 2y + 3x = 8 is
1) translation
2) glide reflection 1) y = 3 x − 4
3) rotation centered at the origin 2
4) reflection through the origin 2) y = 3 x + 4
2
3) y = − 3 x − 2
2
4) y = − 3 x + 2
2

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74 Which diagram shows the construction of the 75 When a quadrilateral is reflected over the line
perpendicular bisector of AB ? y = x, which geometric relationship is not
preserved?
1) congruence
2) orientation
3) parallelism
4) perpendicularity

1) 76 What is an equation of a circle with center (7, −3)

and radius 4?
1) (x − 7) 2 + (y + 3) 2 = 4
2) (x + 7) 2 + (y − 3) 2 = 4
3) (x − 7) 2 + (y + 3) 2 = 16
4) (x + 7) 2 + (y − 3) 2 = 16

2)
77 Line segment AB is shown in the diagram below.

3)

Which two sets of construction marks, labeled I, II,

III, and IV, are part of the construction of the
4) perpendicular bisector of line segment AB?
1) I and II
2) I and III
3) II and III
4) II and IV

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78 A straightedge and compass were used to create the 80 Point M is the midpoint of AB . If the coordinates
construction below. Arc EF was drawn from point of A are (−3, 6) and the coordinates of M are (−5, 2),
B, and arcs with equal radii were drawn from E and what are the coordinates of B?
F. 1) (1, 2)
2) (7, 10)
3) (−4, 4)
4) (−7, −2)

81 What is the volume, in cubic centimeters, of a

cylinder that has a height of 15 cm and a diameter
of 12 cm?
1) 180π
2) 540π
Which statement is false? 3) 675π
1) m∠ABD = m∠DBC 4) 2, 160π
2) 1 (m∠ABC) = m∠ABD
2
3) 2(m∠DBC) = m∠ABC
4) 2(m∠ABC) = m∠CBD
82 The angles of triangle ABC are in the ratio of
8 :3 :4. What is the measure of the smallest angle?
1) 12º
2) 24º
79 In the diagram of KLM below, m∠L = 70, 3) 36º
m∠M = 50, and MK is extended through N. 4) 72º

83 For which polygon does the sum of the measures of

the interior angles equal the sum of the measures of
the exterior angles?
1) hexagon
2) pentagon
3) quadrilateral
4) triangle
What is the measure of ∠LKN ?
1) 60º
2) 120º
3) 180º
4) 300º

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84 What is an equation of the circle shown in the 87 Given that ABCD is a parallelogram, a student
graph below? wrote the proof below to show that a pair of its
opposite angles are congruent.

What is the reason justifying that ∠B ≅ ∠D?

1) Opposite angles in a quadrilateral are
1) (x − 3) 2 + (y − 4) 2 = 25 congruent.
2) (x + 3) 2 + (y + 4) 2 = 25 2) Parallel lines have congruent corresponding
angles.
3) (x − 3) 2 + (y − 4) 2 = 10
3) Corresponding parts of congruent triangles are
4) (x + 3) 2 + (y + 4) 2 = 10 congruent.
4) Alternate interior angles in congruent triangles
are congruent.

85 Which statement is true about every parallelogram?

1) All four sides are congruent.
2) The interior angles are all congruent. 88 Triangle PQR has angles in the ratio of 2 :3 :5.
3) Two pairs of opposite sides are congruent. Which type of triangle is PQR?
4) The diagonals are perpendicular to each other. 1) acute
2) isosceles
3) obtuse
4) right
86 Which equation represents circle O with center
(2, −8) and radius 9?
1) (x + 2) 2 + (y − 8) 2 = 9
2) (x − 2) 2 + (y + 8) 2 = 9
3) (x + 2) 2 + (y − 8) 2 = 81
4) (x − 2) 2 + (y + 8) 2 = 81

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AC CB 92 In the diagram below of circle O, diameter AOB is

89 In ABC and DEF , = . Which
DF FE
perpendicular to chord CD at point E, OA = 6, and
additional information would prove
OE = 2.
ABC ∼ DEF ?
1) AC = DF
2) CB = FE
3) ∠ACB ≅ ∠DFE
4) ∠BAC ≅ ∠EDF

90 In the diagram of ABC shown below, DE  BC .

What is the length of CE ?

1) 4 3
2) 2 3
3) 8 2
4) 4 2
If AB = 10 , AD = 8 , and AE = 12 , what is the
length of EC ?
1) 6
2) 2 93 In the diagram below of ABC , AE ≅ BE ,
3) 3 AF ≅ CF , and CD ≅ BD .
4) 15

91 In RST , m∠R = 58 and m∠S = 73. Which

inequality is true?
1) RT < TS < RS
2) RS < RT < TS
3) RT < RS < TS
4) RS < TS < RT
Point P must be the
1) centroid
2) circumcenter
3) Incenter
4) orthocenter

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94 In the diagram below, ABC ∼ RST . 96 In the diagram below, line p intersects line m and
line n.

Which statement is not true?

1) ∠A ≅ ∠R
2) AB = BC
RS ST
3) AB = ST
BC RS
If m∠1 = 7x and m∠2 = 5x + 30, lines m and n are
AB + BC + AC AB
4) = parallel when x equals
RS + ST + RT RS
1) 12.5
2) 15
3) 87.5
4) 105
95 In the diagram below of circle O, radius OC is 5
cm. Chord AB is 8 cm and is perpendicular to OC
at point P.
97 As shown in the diagram below, AC bisects ∠BAD
and ∠B ≅ ∠D.

Which method could be used to prove

What is the length of OP, in centimeters? ABC ≅ ADC ?
1) 8 1) SSS
2) 2 2) AAA
3) 3 3) SAS
4) 4 4) AAS

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98 Parallelogram ABCD has coordinates A(1, 5) , 101 What is an equation of the line that is perpendicular
B(6, 3) , C(3, −1), and D(−2, 1) . What are the to the line whose equation is y = 3 x − 2 and that
5
coordinates of E, the intersection of diagonals AC
passes through the point (3, −6)?
and BD ?
1) (2, 2) 1) y = 5 x − 11
3
2) (4.5, 1)
3) (3.5, 2) 2) y = − 5 x + 11
3
4) (−1, 3)
3) y = − 5 x − 1
3
4) y = 5 x + 1
3
99 In the diagram of JEA below, m∠JEA = 90 and
m∠EAJ = 48. Line segment MS connects points M
and S on the triangle, such that m∠EMS = 59.
←→
102 In the diagram below of ABC , TV  BC , AT = 5 ,
TB = 7, and AV = 10 .

What is m∠JSM ?
What is the length of VC ?
1) 163
2) 121 1) 3 1
2
3) 42
4) 17 2) 7 1
7
3) 14
4) 24

100 When ABC is dilated by a scale factor of 2, its

image is A ′B ′C ′. Which statement is true?
1) AC ≅ A ′C ′
2) ∠A ≅ ∠A ′
3) perimeter of ABC = perimeter of A ′B ′C ′
4) 2(area of ABC ) = area of A ′B ′C ′

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103 If two distinct planes, A and B, are perpendicular 105 As shown in the diagram below, lines m and n are
to line c, then which statement is true? cut by transversal p.
1) Planes A and B are parallel to each other.
2) Planes A and B are perpendicular to each
other.
3) The intersection of planes A and B is a line
parallel to line c.
4) The intersection of planes A and B is a line
perpendicular to line c.

If m∠1 = 4x + 14 and m∠2 = 8x + 10, lines m and n

are parallel when x equals
104 As shown in the diagram below, a kite needs a 1) 1
vertical and a horizontal support bar attached at 2) 6
opposite corners. The upper edges of the kite are 7 3) 13
inches, the side edges are x inches, and the vertical 4) 17
support bar is (x + 1) inches.

106 In ABC shown below, P is the centroid and

BF = 18 .

What is the measure, in inches, of the vertical

support bar?
1) 23
2) 24 What is the length of BP ?
3) 25 1) 6
2) 9
4) 26
3) 3
4) 12

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107 In circle O, a diameter has endpoints (−5, 4) and 110 In the diagram below, quadrilateral JUMP is
(3, −6). What is the length of the diameter? inscribed in a circle..
1) 2
2) 2 2
3) 10
4) 2 41

108 In the diagram below, MATH is a rhombus with

diagonals AH and MT .

Opposite angles J and M must be

1) right
2) complementary
3) congruent
4) supplementary

111 In the diagram of quadrilateral ABCD, AB  CD,

∠ABC ≅ ∠CDA, and diagonal AC is drawn.
If m∠HAM = 12, what is m∠AMT ?
1) 12
2) 78
3) 84
4) 156

109 What is the length of the line segment whose

endpoints are A(−1, 9) and B(7, 4) ?
1) 61 Which method can be used to prove ABC is
2) 89 congruent to CDA?
3) 205 1) AAS
2) SSA
4) 233
3) SAS
4) SSS

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112 In PQR, ∠PRQ is a right angle and RT is drawn 115 What is the length of AB with endpoints A(−1, 0)
perpendicular to hypotenuse PQ . If PT = x , and B(4, −3) ?
RT = 6 , and TQ = 4x, what is the length of PQ ? 1) 6
1) 9 2) 18
2) 12 3) 34
3) 3 4) 50
4) 15

←→

113 The point (3, −2) is rotated 90º about the origin and 116 As shown in the diagram below, EF intersects
then dilated by a scale factor of 4. What are the planes P, Q, and R.
coordinates of the resulting image?
1) (−12, 8)
2) (12,-8)
3) (8, 12)
4) (−8, −12)

←→
114 The diagram below shows a pair of congruent If EF is perpendicular to planes P and R, which
triangles, with ∠ADB ≅ ∠CDB and statement must be true?
∠ABD ≅ ∠CBD.
1) Plane P is perpendicular to plane Q.
2) Plane R is perpendicular to plane P.
3) Plane P is parallel to plane Q.
4) Plane R is parallel to plane P.

117 When a dilation is performed on a hexagon, which

property of the hexagon will not be preserved in its
Which statement must be true? image?
1) ∠ADB ≅ ∠CBD 1) parallelism
2) ∠ABC ≅ ∠ADC 2) orientation
3) AB ≅ CD 3) length of sides
4) AD ≅ CD 4) measure of angles

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Geometry Multiple Choice Regents Exam Questions
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118 In the diagram below of DAE and BCE , AB and 121 Triangle ABC is graphed on the set of axes below.
CD intersect at E, such that AE ≅ CE and
∠BCE ≅ ∠DAE .

Which transformation produces an image that is

similar to, but not congruent to, ABC ?
Triangle DAE can be proved congruent to triangle
1) T 2, 3
BCE by
1) ASA 2) D2
2) SAS 3) ry= x
3) SSS 4) R 90
4) HL

119 Which statement is the negation of “Two is a prime 122 What is the equation of the line that passes through
number” and what is the truth value of the the point (−9, 6) and is perpendicular to the line
negation? y = 3x − 5?
1) Two is not a prime number; false 1) y = 3x + 21
2) Two is not a prime number; true
2) y = − 1 x − 3
3) A prime number is two; false 3
4) A prime number is two; true 3) y = 3x + 33
4) y = − 1 x + 3
3

120 How many points are both 4 units from the origin
and also 2 units from the line y = 4?
1) 1 123 Which equation of a circle will have a graph that
2) 2 lies entirely in the first quadrant?
3) 3 1) (x − 4) 2 + (y − 5) 2 = 9
4) 4
2) (x + 4) 2 + (y + 5) 2 = 9
3) (x + 4) 2 + (y + 5) 2 = 25
4) (x − 5) 2 + (y − 4) 2 = 25

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124 Which equation represents the line that is 127 As shown in the diagram of ACD below, B is a
perpendicular to 2y = x + 2 and passes through the point on AC and DB is drawn.
point (4, 3)?
1) y = 1 x − 5
2
2) y = 1 x + 1
2
3) y = −2x + 11
4) y = −2x − 5

If m∠A = 66, m∠CDB = 18, and m∠C = 24, what is

the longest side of ABD?
125 In the diagram below of right triangle ABC, CD is 1) AB
the altitude to hypotenuse AB , CB = 6, and AD = 5 . 2) DC
3) AD
4) BD

128 A packing carton in the shape of a triangular prism

is shown in the diagram below.

What is the length of BD ?

1) 5
2) 9
3) 3
4) 4

126 The two lines represented by the equations below What is the volume, in cubic inches, of this carton?
are graphed on a coordinate plane. 1) 20
x + 6y = 12 2) 60
3) 120
3(x − 2) = −y − 4
4) 240
Which statement best describes the two lines?
1) The lines are parallel.
2) The lines are the same line.
3) The lines are perpendicular.
4) The lines intersect at an angle other than 90°.

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129 Which graph represents a circle whose equation is 131 Which equation represents a line that is parallel to
(x + 2) 2 + y 2 = 16? the line whose equation is y = 3 x − 3 and passes
2
through the point (1, 2)?
1) y = 3 x + 1
2 2
2) y = 2 x + 4
3 3
3
3) y = x − 2
2
1)
4) y = − 2 x + 8
3 3

132 What is the length of the line segment whose

endpoints are (1, −4) and (9, 2)?
2) 1) 5
2) 2 17
3) 10
4) 2 26

133 In the diagram below, point P is the centroid of

3)
ABC .

4)

If PM = 2x + 5 and BP = 7x + 4 , what is the length

130 If the vertex angles of two isosceles triangles are
of PM ?
congruent, then the triangles must be
1) 9
1) acute
2) 2
2) congruent
3) 18
3) right
4) 27
4) similar

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Geometry Multiple Choice Regents Exam Questions
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134 Which diagram represents a correct construction of 135 In the diagram below, DE joins the midpoints of
equilateral ABC , given side AB ? two sides of ABC .

1) Which statement is not true?

1) CE = 1 CB
2
2) DE = 1 AB
2
3) area of CDE = 1 area of CAB
2
4) perimeter of CDE = 1 perimeter of CAB
2

2)
136 Point P lies on line m. Point P is also included in
distinct planes Q, R, S, and T. At most, how many
of these planes could be perpendicular to line m?
1) 1
2) 2
3) 3
3) 4) 4

137 A line segment has endpoints (4, 7) and (1, 11).

What is the length of the segment?
1) 5
2) 7
3) 16
4) 25
4)

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138 In the diagram below, parallelogram ABCD has 140 A student wrote the sentence “4 is an odd integer.”
diagonals AC and BD that intersect at point E. What is the negation of this sentence and the truth
value of the negation?
1) 3 is an odd integer; true
2) 4 is not an odd integer; true
3) 4 is not an even integer; false
4) 4 is an even integer; false

Which expression is not always true? 141 What is the image of the point (2, −3) after the
1) ∠DAE ≅ ∠BCE transformation r y − axis?
2) ∠DEC ≅ ∠BEA
1) (2, 3)
3) AC ≅ DB
2) (−2, −3)
4) DE ≅ EB
3) (−2, 3)
4) (−3, 2)

139 In the diagram below of ACE , medians AD , EB ,

and CF intersect at G. The length of FG is 12 cm.
142 The equation of line k is y = 1 x − 2. The equation
3
of line m is −2x + 6y = 18. Lines k and m are
1) parallel
2) perpendicular
3) the same line
4) neither parallel nor perpendicular

143 Which set of numbers does not represent the sides

of a right triangle?
1) {6, 8, 10}
What is the length, in centimeters, of GC ? 2) {8, 15, 17}
1) 24 3) {8, 24, 25}
2) 12 4) {15, 36, 39}
3) 6
4) 4

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Geometry Multiple Choice Regents Exam Questions
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←→
144 Which graph represents a circle with the equation
145 The diagram below shows the construction of AB
(x − 3) 2 + (y + 1) 2 = 4? ←
→
through point P parallel to CD .

1)

Which theorem justifies this method of

construction?
1) If two lines in a plane are perpendicular to a
transversal at different points, then the lines are
parallel.
2) If two lines in a plane are cut by a transversal
2) to form congruent corresponding angles, then
the lines are parallel.
3) If two lines in a plane are cut by a transversal
to form congruent alternate interior angles,
then the lines are parallel.
4) If two lines in a plane are cut by a transversal
to form congruent alternate exterior angles,
then the lines are parallel.

3)

146 Plane R is perpendicular to line k and plane D is

perpendicular to line k. Which statement is
correct?
1) Plane R is perpendicular to plane D.
2) Plane R is parallel to plane D.
3) Plane R intersects plane D.
4) 4) Plane R bisects plane D.

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Geometry Multiple Choice Regents Exam Questions
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147 In the diagram below of isosceles trapezoid ABCD, 149 What is the image of the point (−5, 2) under the
AB = CD = 25 , AD = 26 , and BC = 12 . translation T 3, −4 ?
1) (−9, 5)
2) (−8, 6)
3) (−2, −2)
4) (−15, −8)

150 The equation of a circle with its center at (−3, 5)

and a radius of 4 is
What is the length of an altitude of the trapezoid?
1) (x + 3) 2 + (y − 5) 2 = 4
1) 7
2) 14 2) (x − 3) 2 + (y + 5) 2 = 4
3) 19 3) (x + 3) 2 + (y − 5) 2 = 16
4) 24
4) (x − 3) 2 + (y + 5) 2 = 16

148 In the diagram below of PAO, AP is tangent to

circle O at point A, OB = 7, and BP = 18 . 151 In the diagram below of circle O, chords AB and
CD intersect at E.

What is the length of AP ?

1) 10 If m∠AEC = 34 and mAC = 50, what is mDB ?
2) 12 1) 16
3) 17 2) 18
4) 24 3) 68
4) 118

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152 Triangle ABC has vertices A(0, 0) , B(3, 2) , and 155 The statement "x is a multiple of 3, and x is an even
C(0, 4). The triangle may be classified as integer" is true when x is equal to
1) equilateral 1) 9
2) isosceles 2) 8
3) right 3) 3
4) scalene 4) 6

153 The sum of the interior angles of a polygon of n 156 Lines m and n intersect at point A. Line k is
sides is perpendicular to both lines m and n at point A.
1) 360 Which statement must be true?
1) Lines m, n, and k are in the same plane.
2) 360 2) Lines m and n are in two different planes.
n
3) (n − 2) ⋅ 180 3) Lines m and n are perpendicular to each other.
(n − 2) ⋅ 180 4) Line k is perpendicular to the plane containing
4) lines m and n.
n

157 The diagonals of a quadrilateral are congruent but

154 In the diagram of ABC shown below, D is the
do not bisect each other. This quadrilateral is
midpoint of AB , E is the midpoint of BC , and F is 1) an isosceles trapezoid
the midpoint of AC . 2) a parallelogram
3) a rectangle
4) a rhombus

158 Lines a and b intersect at point P. Line c passes

through P and is perpendicular to the plane
containing lines a and b. Which statement must be
true?
If AB = 20 , BC = 12 , and AC = 16 , what is the 1) Lines a, b, and c are coplanar.
perimeter of trapezoid ABEF? 2) Line a is perpendicular to line b.
1) 24 3) Line c is perpendicular to both line a and line
2) 36 b.
3) 40 4) Line c is perpendicular to line a or line b, but
4) 44 not both.

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159 The volume, in cubic centimeters, of a sphere 162 Point A lies in plane B. How many lines can be
whose diameter is 6 centimeters is drawn perpendicular to plane B through point A?
1) 12π
1) one
2) 36π
2) two
3) 48π
3) zero
4) 288π
4) infinite

160 In ABC , AB = 5 feet and BC = 3 feet. Which

163 As shown in the diagram below of ABC , a
inequality represents all possible values for the
compass is used to find points D and E, equidistant
length of AC , in feet? from point A. Next, the compass is used to find
1) 2 ≤ AC ≤ 8 point F, equidistant from points D and E. Finally, a
2) 2 < AC < 8 
→
3) 3 ≤ AC ≤ 7 straightedge is used to draw AF . Then, point G,

→
4) 3 < AC < 7 the intersection of AF and side BC of ABC , is
labeled.

161 In circle O shown below, diameter DB is

perpendicular to chord AC at E.

Which statement must be true?

→
1) AF bisects side BC
→
If DB = 34 , AC = 30 , and DE > BE , what is the 2) AF bisects ∠BAC
→
length of BE ? 3) AF ⊥ BC
1) 8 4) ABG ∼ ACG
2) 9
3) 16
4) 25

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164 A circle has the equation (x − 2) 2 + (y + 3) 2 = 36. 167 The diagram below shows a rectangular prism.
What are the coordinates of its center and the
length of its radius?
1) (−2, 3) and 6
2) (2, −3) and 6
3) (−2, 3) and 36
4) (2, −3) and 36

165 Which compound statement is true?

1) A triangle has three sides and a quadrilateral Which pair of edges are segments of lines that are
has five sides. coplanar?
2) A triangle has three sides if and only if a 1) AB and DH
quadrilateral has five sides.
2) AE and DC
3) If a triangle has three sides, then a quadrilateral
has five sides. 3) BC and EH
4) A triangle has three sides or a quadrilateral has 4) CG and EF
five sides.

168 As shown in the diagram below, FJ is contained in

166 The diagram below represents a rectangular solid.
plane R, BC and DE are contained in plane S, and
FJ , BC , and DE intersect at A.

Which statement must be true?

1) EH and BC are coplanar
2) FG and AB are coplanar Which fact is sufficient to show that planes R and
3) EH and AD are skew S are perpendicular?
4) FG and CG are skew
1) FA ⊥ DE
2) AD ⊥ AF
3) BC ⊥ FJ
4) DE ⊥ BC

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Geometry Multiple Choice Regents Exam Questions

169 Based on the construction below, which conclusion 171 The solution of the system of equations y = x 2 − 2
is not always true? and y = x is
1) (1, 1) and (−2, −2)
2) (2, 2) and (−1, −1)
3) (1, 1) and (2, 2)
4) (−2, −2) and (−1, −1)

172 In circle O, diameter AB intersects chord CD at E.

If CE = ED, then ∠CEA is which type of angle?
1) straight
2) obtuse
3) acute
1) AB ⊥ CD 4) right
2) AB = CD
3) AE = EB
4) CE = DE
173 What is the equation for circle O shown in the
graph below?

170 Circle O with ∠AOC and ∠ABC is shown in the

diagram below.

What is the ratio of m∠AOC to m∠ABC ?

1) 1 : 1 1) (x − 3) 2 + (y + 1) 2 = 6
2) 2 : 1 2) (x + 3) 2 + (y − 1) 2 = 6
3) 3 : 1
4) 1 : 2 3) (x − 3) 2 + (y + 1) 2 = 9
4) (x + 3) 2 + (y − 1) 2 = 9

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←
→
174 In the diagram below, RCBT and ABC are shown 177 In the diagram below, diameter AB bisects chord
with m∠A = 60 and m∠ABT = 125. CD at point E in circle F.

What is m∠ACR?
1) 125
2) 115 If AE = 2 and FB = 17 , then the length of CE is
3) 65 1) 7
4) 55 2) 8
3) 15
4) 16

175 A rectangular prism has a base with a length of 25,

a width of 9, and a height of 12. A second prism
has a square base with a side of 15. If the volumes 178 As shown below, the medians of ABC intersect at
of the two prisms are equal, what is the height of D.
the second prism?
1) 6
2) 8
3) 12
4) 15

176 Triangle ABC is similar to triangle DEF. The

lengths of the sides of ABC are 5, 8, and 11. If the length of BE is 12, what is the length of BD ?
What is the length of the shortest side of DEF if 1) 8
its perimeter is 60? 2) 9
1) 10 3) 3
2) 12.5 4) 4
3) 20
4) 27.5

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179 If ABC and its image, A ′B ′C ′, are graphed on a 182 Point A is on line m. How many distinct planes
set of axes, ABC ≅ A ′B ′C ′ under each will be perpendicular to line m and pass through
transformation except point A?
1) D 2 1) one
2) R 90° 2) two
3) r y = x 3) zero
4) infinite
4) T (−2, 3)

183 Which equation represents the circle whose center

180 If ABC ≅ JKL ≅ RST , then BC must be is (−5, 3) and that passes through the point (−1, 3)?
congruent to 1) (x + 1) 2 + (y − 3) 2 = 16
1) JL 2) (x − 1) 2 + (y + 3) 2 = 16
2) JK 3) (x + 5) 2 + (y − 3) 2 = 16
3) ST 4) (x − 5) 2 + (y + 3) 2 = 16
4) RS

184 Given: ABD, BC is the perpendicular bisector of

181 In ABC , D is the midpoint of AB and E is the
AD
midpoint of BC . If AC = 3x − 15 and DE = 6 , what
is the value of x?

Which statement can not always be proven?

1) AC ≅ DC
1) 6 2) BC ≅ CD
2) 7 3) ∠ACB ≅ ∠DCB
3) 9 4) ABC ≅ DBC
4) 12

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185 Triangle ABC has vertices A(0, 0) , B(6, 8) , and 189 In the diagram of UVW below, A is the midpoint
C(8, 4). Which equation represents the of UV , B is the midpoint of UW , C is the midpoint
perpendicular bisector of BC ? of VW , and AB and AC are drawn.
1) y = 2x − 6
2) y = −2x + 4
3) y = 1 x + 5
2 2
4) y = − 1 x + 19
2 2

186 What is the equation of a line passing through the

point (6, 1) and parallel to the line whose equation
is 3x = 2y + 4?
1) y = − 2 x + 5 If VW = 7x − 3 and AB = 3x + 1 , what is the length
3
of VC ?
2) y = − 2 x − 3 1) 5
3
2) 13
3) y = 3 x − 8 3) 16
2
4) 32
4) y = 3 x − 5
2

190 What are the coordinates of A ′, the image of

187 Triangle ABC has the coordinates A(1, 2) , B(5, 2) , A(−3, 4) , after a rotation of 180º about the origin?
and C(5, 5). Triangle ABC is rotated 180° about the 1) (4, −3)
origin to form triangle A' B' C' . Triangle A' B' C' is 2) (−4, −3)
1) acute 3) (3, 4)
2) isosceles 4) (3, −4)
3) obtuse
4) right

191 When the system of equations y + 2 = (x − 4) 2 and

2x + y − 6 = 0 is solved graphically, the solution is
188 A circle with the equation (x + 6) 2 + (y − 7) 2 = 64 1) (−4, −2) and (−2, 2)
does not include points in Quadrant 2) (4, −2) and (2, 2)
1) I 3) (−4, 2) and (−6, 6)
2) II
4) (4, 2) and (6, 6)
3) III
4) IV

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192 A rectangular right prism is shown in the diagram 195 In trapezoid RSTV with bases RS and VT , diagonals
below.
RT and SV intersect at Q.

If trapezoid RSTV is not isosceles, which triangle is

equal in area to RSV ?
1) RQV
Which pair of edges are not coplanar? 2) RST
1) BF and CG 3) RVT
2) BF and DH 4) SVT
3) EF and CD
4) EF and BC

196 In the diagram below, four pairs of triangles are

shown. Congruent corresponding parts are labeled
in each pair.
193 The sides of a triangle are 8, 12, and 15. The
longest side of a similar triangle is 18. What is the
ratio of the perimeter of the smaller triangle to the
perimeter of the larger triangle?
1) 2 :3
2) 4 :9
3) 5 :6
4) 25 :36

194 Which equation represents a line that is parallel to

the line whose equation is 3x − 2y = 7?
1) y = − 3 x + 5
2
2) y = − 2 x + 4
3 Using only the information given in the diagrams,
which pair of triangles can not be proven
3) y = 3 x − 5
2 congruent?
1) A
4) y = 2 x − 4
3 2) B
3) C
4) D

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197 The equations x 2 + y 2 = 25 and y = 5 are graphed 200 In the diagram below, XYV ≅ TSV .
on a set of axes. What is the solution of this
system?
1) (0, 0)
2) (5, 0)
3) (0, 5)
4) (5, 5)

198 If the vertices of ABC are A(−2, 4) , B(−2, 8) , and Which statement can not be proven?
C(−5, 6), then ABC is classified as 1) ∠XVY ≅ ∠TVS
1) right 2) ∠VYX ≅ ∠VUT
2) scalene 3) XY ≅ TS
3) isosceles 4) YV ≅ SV
4) equilateral

←→
201 Points A(5, 3) and B(7, 6) lie on AB . Points C(6, 4)
199 Triangle ABC shown below is a right triangle with ←
→
altitude AD drawn to the hypotenuse BC . and D(9, 0) lie on CD . Which statement is true?
←→ ←
→
1) AB  CD
←→ ←
→
2) AB ⊥ CD
←→ ←
→
3) AB and CD are the same line.
←→ ←
→
4) AB and CD intersect, but are not
perpendicular.

If BD = 2 and DC = 10 , what is the length of AB ?

1) 2 2 202 Square ABCD has vertices A(−2, −3) , B(4, −1) ,
C(2, 5), and D(−4, 3) . What is the length of a side
2) 2 5
of the square?
3) 2 6
1) 2 5
4) 2 30
2) 2 10
3) 4 5
4) 10 2

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203 As shown in the diagram below, CD is a median of 206 As shown in the diagram below, when right
ABC . triangle DAB is reflected over the x-axis, its image
is triangle DCB.

Which statement is always true?

1) AD ≅ DB
2) AC ≅ AD
3) ∠ACD ≅ ∠CDB
4) ∠BCD ≅ ∠ACD

Which statement justifies why AB ≅ CB?

204 Line  passes through the point (5, 3) and is parallel
1) Distance is preserved under reflection.
to line k whose equation is 5x + y = 6. An equation
2) Orientation is preserved under reflection.
of line  is 3) Points on the line of reflection remain
1) y = 1 x + 2 invariant.
5 4) Right angles remain congruent under
2) y = −5x + 28 reflection.
3) y = 1 x − 2
5
4) y = −5x − 28
207 In a coordinate plane, the locus of points 5 units
from the x-axis is the
1) lines x = 5 and x = −5
205 What is the perimeter of a square whose diagonal is 2) lines y = 5 and y = −5
3 2? 3) line x = 5, only
1) 18 4) line y = 5, only
2) 12
3) 9
4) 6

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208 The diagram below shows the construction of line 210 What is the slope of the line perpendicular to the
m, parallel to line , through point P. line represented by the equation 2x + 4y = 12?
1) −2
2) 2
3) − 1
2
4) 1
2

211 Trapezoid QRST is graphed on the set of axes

below.
Which theorem was used to justify this
construction?
1) If two lines are cut by a transversal and the
alternate interior angles are congruent, the lines
are parallel.
2) If two lines are cut by a transversal and the
interior angles on the same side are
supplementary, the lines are parallel.
3) If two lines are perpendicular to the same line,
they are parallel.
4) If two lines are cut by a transversal and the
corresponding angles are congruent, they are
parallel.
Under which transformation will there be no
invariant points?
1) r y = 0
2) rx = 0
209 What is the converse of “If an angle measures 90
degrees, then it is a right angle”? 3) r (0, 0)
1) If an angle is a right angle, then it measures 90 4) ry= x
degrees.
2) An angle is a right angle if it measures 90
degrees.
3) If an angle is not a right angle, then it does not
measure 90 degrees. 212 If MNP ≅ VWX and PM is the shortest side of
4) If an angle does not measure 90 degrees, then it MNP, what is the shortest side of VWX ?
is not a right angle. 1) XV
2) WX
3) VW
4) NP

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→
213 As shown in the diagram below, FD and CB 215 The diagram below shows ABD, with ABC ,
intersect at point A and ET is perpendicular to both BE ⊥ AD , and ∠EBD ≅ ∠CBD.
FD and CB at A.

If m∠ABE = 52, what is m∠D?

Which statement is not true?
1) 26
1) ET is perpendicular to plane BAD. 2) 38
2) ET is perpendicular to plane FAB. 3) 52
3) ET is perpendicular to plane CAD. 4) 64
4) ET is perpendicular to plane BAT.

216 As shown in the diagram of rectangle ABCD

214 In circle R shown below, diameter DE is below, diagonals AC and BD intersect at E.
perpendicular to chord ST at point L.

If AE = x + 2 and BD = 4x − 16 , then the length of

AC is
1) 6
2) 10
3) 12
4) 24

Which statement is not always true?

1) SL ≅ TL
2) RS = DR
3) RL ≅ LE
4) (DL)(LE) = (SL)(LT)

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217 How many points in the coordinate plane are 3 220 In the diagram of ABC below, AB is extended to
units from the origin and also equidistant from both point D.
the x-axis and the y-axis?
1) 1
2) 2
3) 8
4) 4

218 The measure of an interior angle of a regular

If m∠CAB = x + 40, m∠ACB = 3x + 10,
polygon is 120°. How many sides does the polygon
m∠CBD = 6x, what is m∠CAB?
have?
1) 13
1) 5
2) 25
2) 6
3) 53
3) 3
4) 65
4) 4

221 Which set of equations represents two circles that

219 Triangle ABC is shown in the diagram below.
have the same center?
1) x 2 + (y + 4) 2 = 16 and (x + 4) 2 + y 2 = 16
2) (x + 3) 2 + (y − 3) 2 = 16 and
(x − 3) 2 + (y + 3) 2 = 25
3) (x − 7) 2 + (y − 2) 2 = 16 and
(x + 7) 2 + (y + 2) 2 = 25
4) (x − 2) 2 + (y − 5) 2 = 16 and
If DE joins the midpoints of ADC and AEB , which (x − 2) 2 + (y − 5) 2 = 25
statement is not true?
1) DE = 1 CB
2
2) DE  CB 222 In ABC , ∠A ≅ ∠B and ∠C is an obtuse angle.
Which statement is true?
3) AD = DE
DC CB 1) AC ≅ AB and BC is the longest side.
4) ABC ∼ AED 2) AC ≅ BC and AB is the longest side.
3) AC ≅ AB and BC is the shortest side.
4) AC ≅ BC and AB is the shortest side.

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227 In the diagram below of quadrilateral ABCD, E and

223 The equation of a line is y = 2 x + 5. What is an
3 F are points on AB and CD, respectively, BE ≅ DF ,
equation of the line that is perpendicular to the
and AE ≅ CF .
given line and that passes through the point (4, 2)?
1) y = 2 x − 2
3 3
2) y = 3 x − 4
2
3) y = − 3 x + 7
2
4) y = − 3 x + 8
2

Which conclusion can be proven?

1) ED ≅ FB
224 The lateral area of a right circular cone is equal to 2) AB ≅ CD
3) ∠A ≅ ∠C
120π cm 2 . If the base of the cone has a diameter
4) ∠AED ≅ ∠CFB
of 24 cm, what is the length of the slant height, in
centimeters?
1) 2.5
2) 5
3) 10 228 In the diagram below of right triangle ABC, altitude
4) 15.7 CD is drawn to hypotenuse AB .

225 Line segment AB is a diameter of circle O whose

center has coordinates (6, 8). What are the
coordinates of point B if the coordinates of point A
are (4, 2)?
1) (1, 3) If AD = 3 and DB = 12 , what is the length of
2) (5, 5) altitude CD?
3) (8, 14) 1) 6
4) (10, 10) 2) 6 5
3) 3
4) 3 5

226 If line  is perpendicular to distinct planes P and Q,

then planes P and Q
1) are parallel
2) contain line 
3) are perpendicular
4) intersect, but are not perpendicular

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229 Which diagram shows the construction of a 45° 230 Which equation represents circle O shown in the
angle? graph below?

1)

1) x 2 + (y − 2) 2 = 10
2) x 2 + (y + 2) 2 = 10
3) x 2 + (y − 2) 2 = 25
2) 4) x 2 + (y + 2) 2 = 25

231 What are the coordinates of the center and the

length of the radius of the circle whose equation is
3) (x + 1) 2 + (y − 5) 2 = 16?
1) (1, −5) and 16
2) (−1, 5) and 16
3) (1, −5) and 4
4) (−1, 5) and 4

232 Which set of numbers could not represent the

lengths of the sides of a right triangle?
4) 1) {1, 3, 10 }
2) {2, 3, 4}
3) {3, 4, 5}
4) {8, 15, 17}

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233 Two prisms have equal heights and equal volumes. 236 As shown in the diagram below, a landscaper uses
The base of one is a pentagon and the base of the a cylindrical lawn roller on a lawn. The roller has a
other is a square. If the area of the pentagonal base radius of 9 inches and a width of 42 inches.
is 36 square inches, how many inches are in the
length of each side of the square base?
1) 6
2) 9
3) 24
4) 36

234 Chords AB and CD intersect at point E in a circle

with center at O. If AE = 8 , AB = 20 , and DE = 16 ,
what is the length of CE ?
1) 6 To the nearest square inch, the area the roller
2) 9 covers in one complete rotation is
3) 10 1) 2,374
4) 12 2) 2,375
3) 10,682
4) 10,688

235 In the diagram below, AC and BC are tangent to

circle O at A and B, respectively, from external
point C. 237 In the diagram of ABC below, medians AD and
BE intersect at point F.

If AF = 6 , what is the length of FD ?

If m∠ACB = 38, what is m∠AOB?
1) 6
1) 71
2) 2
2) 104
3) 3
3) 142
4) 9
4) 161

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238 How many points are 5 units from a line and also 241 A student wrote the following equations:
equidistant from two points on the line? 3y + 6 = 2x
1) 1
2y − 3x = 6
2) 2
The lines represented by these equations are
3) 3
1) parallel
4) 0
2) the same line
3) perpendicular
4) intersecting, but not perpendicular

239 Lines m and n are in plane A. What is the converse

of the statement “If lines m and n are parallel, then
lines m and n do not intersect”? 242 In right triangle ABC shown in the diagram below,
1) If lines m and n are not parallel, then lines m
altitude BD is drawn to hypotenuse AC , CD = 12,
and n intersect.
and AD = 3 .
2) If lines m and n are not parallel, then lines m
and n do not intersect
3) If lines m and n intersect, then lines m and n are
not parallel.
4) If lines m and n do not intersect, then lines m
and n are parallel.

240 In the diagram of circle O below, chord CD is

parallel to diameter AOB and mCD = 110.

What is the length of AB ?

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

243 The coordinates of point P are (7, 1). What are the
What is mDB ? coordinates of the image of P after R 90° about the
1) 35 origin?
2) 55 1) (1, 7)
3) 70 2) (−7, −1)
4) 110 3) (1, −7)
4) (−1, 7)

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244 In the diagram below, AC and AD are tangent to 247 Given the statement: One is a prime number.
What is the negation and the truth value of the
circle B at points C and D, respectively, and BC ,
negation?
BD , and BA are drawn. 1) One is not a prime number; true
2) One is not a prime number; false
3) One is a composite number; true
4) One is a composite number; false

248 What is the equation of circle O shown in the

diagram below?

If AC = 12 and AB = 15 , what is the length of BD ?

1) 5.5
2) 9
3) 12
4) 18

245 Which quadrilateral does not always have

congruent diagonals?
1) isosceles trapezoid
2) rectangle
3) rhombus
4) square
1) (x + 4) 2 + (y − 1) 2 = 3
2) (x − 4) 2 + (y + 1) 2 = 3
3) (x + 4) 2 + (y − 1) 2 = 9
246 What is an equation of the circle with center (−5, 4) 4) (x − 4) 2 + (y + 1) 2 = 9
and a radius of 7?
1) (x − 5) 2 + (y + 4) 2 = 14
2) (x − 5) 2 + (y + 4) 2 = 49
3) (x + 5) 2 + (y − 4) 2 = 14 249 What is the measure of the largest exterior angle
that any regular polygon can have?
4) (x + 5) 2 + (y − 4) 2 = 49 1) 60º
2) 90º
3) 120º
4) 360º

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250 Plane A and plane B are two distinct planes that are 253 The bases of a right triangular prism are ABC and
both perpendicular to line . Which statement DEF . Angles A and D are right angles, AB = 6 ,
about planes A and B is true? AC = 8 , and AD = 12 . What is the length of edge
BE ?
1) Planes A and B have a common edge, which
1) 10
forms a line.
2) 12
2) Planes A and B are perpendicular to each 3) 14
other. 4) 16
3) Planes A and B intersect each other at exactly
one point.
4) Planes A and B are parallel to each other.
254 In the diagram below of circle O, PAC and PBD
are secants.

251 If distinct planes R and S are both perpendicular to

line , which statement must always be true?
1) Plane R is parallel to plane S.
2) Plane R is perpendicular to plane S.
3) Planes R and S and line  are all parallel.
4) The intersection of planes R and S is
perpendicular to line . If mCD = 70 and mAB = 20, what is the degree
measure of ∠P?
1) 25
2) 35
3) 45
252 Which equation represents circle A shown in the
4) 50
diagram below?

255 What is the difference between the sum of the

measures of the interior angles of a regular
pentagon and the sum of the measures of the
exterior angles of a regular pentagon?
1) 36
2) 72
3) 108
1) (x − 4) 2 + (y − 1) 2 = 3 4) 180

2) (x + 4) 2 + (y + 1) 2 = 3
3) (x − 4) 2 + (y − 1) 2 = 9
4) (x + 4) 2 + (y + 1) 2 = 9

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256 Consider the relationship between the two 258 Lines p and q are intersected by line r, as shown
statements below. below.
If 16 + 9 ≠ 4 + 3, then 5 ≠ 4 + 3
If 16 + 9 = 4 + 3, then 5 = 4 + 3
These statements are
1) inverses
2) converses
3) contrapositives
4) biconditionals

If m∠1 = 7x − 36 and m∠2 = 5x + 12, for which

value of x would p  q?
257 As shown in the diagram below, a right pyramid 1) 17
has a square base, ABCD, and EF is the slant 2) 24
height. 3) 83
4) 97

←→ ←→ ←
→
259 Transversal EF intersects AB and CD , as shown in
the diagram below.

Which statement is not true?

1) EA ≅ EC
2) EB ≅ EF
3) AEB ≅ BEC Which statement could always be used to prove
4) CED is isosceles ←→ ←
→
AB  CD ?
1) ∠2 ≅ ∠4
2) ∠7 ≅ ∠8
3) ∠3 and ∠6 are supplementary
4) ∠1 and ∠5 are supplementary

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260 In the diagram of trapezoid ABCD below, AB  DC , 262 Which graph represents a circle whose equation is
AD ≅ BC , m∠A = 4x + 20, and m∠C = 3x − 15. x 2 + (y − 1) 2 = 9?

What is m∠D? 1)
1) 25
2) 35
3) 60
4) 90

261 In the diagram below, under which transformation 2)

is A ′B ′C ′ the image of ABC ?

3)

1) D2
2) r x-axis
3) r y-axis 4)
4) (x, y) → (x − 2, y)

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263 In ABC , m∠A = 3x + 1, m∠B = 4x − 17, and 267 The diagram below shows the construction of an
m∠C = 5x − 20. Which type of triangle is ABC ? equilateral triangle.
1) right
2) scalene
3) isosceles
4) equilateral

264 For which measures of the sides of ABC is angle Which statement justifies this construction?
B the largest angle of the triangle? 1) ∠A + ∠B + ∠C = 180
1) AB = 2, BC = 6, AC = 7 2) m∠A = m∠B = m∠C
2) AB = 6, BC = 12, AC = 8 3) AB = AC = BC
3) AB = 16, BC = 9, AC = 10 4) AB + BC > AC
4) AB = 18, BC = 14, AC = 5

268 How many common tangent lines can be drawn to

265 In ABC , m∠A = 60, m∠B = 80, and m∠C = 40. the circles shown below?
Which inequality is true?
1) AB > BC
2) AC > BC
3) AC < BA
4) BC < BA

266 The midpoint of AB is M(4, 2). If the coordinates

of A are (6, −4), what are the coordinates of B?
1) 1
1) (1, −3)
2) 2
2) (2, 8)
3) 3
3) (5, −1) 4) 4
4) (14, 0)

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269 Which graph represents a circle whose equation is 271 In ABC shown below, L is the midpoint of BC , M
x 2 + (y − 2) 2 = 4? is the midpoint of AB , and N is the midpoint of AC .

1)

If MN = 8, ML = 5, and NL = 6, the perimeter of

trapezoid BMNC is
1) 35
2) 31
3) 28
4) 26
2)

272 What are the coordinates of the center of a circle if

the endpoints of its diameter are A(8, −4) and
B(−3, 2) ?
1) (2.5, 1)
2) (2.5, −1)
3)
3) (5.5, −3)
4) (5.5, 3)

273 What is the equation of the circle with its center at

(−1, 2) and that passes through the point (1, 2)?
4) 1) (x + 1) 2 + (y − 2) 2 = 4
2) (x − 1) 2 + (y + 2) 2 = 4
3) (x + 1) 2 + (y − 2) 2 = 2
270 The volume of a sphere is approximately 44.6022 4) (x − 1) 2 + (y + 2) 2 = 2
cubic centimeters. What is the radius of the sphere,
to the nearest tenth of a centimeter?
1) 2.2
2) 3.3
3) 4.4
4) 4.7

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274 Which graph could be used to find the solution to 275 In circle O shown in the diagram below, chords AB
the following system of equations?
and CD are parallel.
y = (x + 3) 2 − 1
x+y=2

1)

If mAB = 104 and mCD = 168, what is mBD ?

1) 38
2) 44
3) 88
4) 96

2)
276 In a park, two straight paths intersect. The city
wants to install lampposts that are both equidistant
from each path and also 15 feet from the
intersection of the paths. How many lampposts are
needed?
1) 1
2) 2
3) 3
3) 4) 4

277 The equation of a circle is (x − 2) 2 + (y + 5) 2 = 32.

What are the coordinates of the center of this circle
and the length of its radius?
1) (−2, 5) and 16
4) 2) (2, −5) and 16
3) (−2, 5) and 4 2
4) (2, −5) and 4 2

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Geometry Multiple Choice Regents Exam Questions

278 Line segment AB has endpoints A(2, −3) and 281 In ABC , m∠A = x, m∠B = 2x + 2, and
B(−4, 6) . What are the coordinates of the midpoint m∠C = 3x + 4. What is the value of x?
1) 29
of AB ?
2) 31
1) (−2, 3)
3) 59
 
2)  −1, 1 1  4) 61
 2
3) (−1, 3)
 
4)  3, 4 1 
 2 282 In the diagram below of regular pentagon ABCDE,
EB is drawn.

279 In the diagram below of parallelogram ABCD with

diagonals AC and BD , m∠1 = 45 and
m∠DCB = 120.

What is the measure of ∠AEB?

1) 36º
2) 54º
What is the measure of ∠2?
3) 72º
1) 15º
4) 108º
2) 30º
3) 45º
4) 60º

283 The lateral faces of a regular pyramid are

composed of
1) squares
280 A quadrilateral whose diagonals bisect each other
2) rectangles
and are perpendicular is a
3) congruent right triangles
1) rhombus
4) congruent isosceles triangles
2) rectangle
3) trapezoid
4) parallelogram

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284 In the diagram below, the length of the legs AC and 287 The diagram below shows a pennant in the shape of
an isosceles triangle. The equal sides each measure
BC of right triangle ABC are 6 cm and 8 cm,
13, the altitude is x + 7, and the base is 2x.
respectively. Altitude CD is drawn to the
hypotenuse of ABC .

What is the length of AD to the nearest tenth of a What is the length of the base?
centimeter? 1) 5
1) 3.6 2) 10
2) 6.0 3) 12
3) 6.4 4) 24
4) 4.0

288 In the diagram below of ABC , medians AD , BE ,

285 Point P is on line m. What is the total number of
and CF intersect at G.
planes that are perpendicular to line m and pass
through point P?
1) 1
2) 2
3) 0
4) infinite

If CF = 24, what is the length of FG ?

286 A transversal intersects two lines. Which condition 1) 8
would always make the two lines parallel? 2) 10
1) Vertical angles are congruent. 3) 12
2) Alternate interior angles are congruent. 4) 16
3) Corresponding angles are supplementary.
4) Same-side interior angles are complementary.

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289 Given ABC with base AFEDC , median BF , 292 In the diagram of ABC below, Jose found
centroid P by constructing the three medians. He
altitude BD , and BE bisects ∠ABC , which
conclusion is valid? measured CF and found it to be 6 inches.

If PF = x , which equation can be used to find x?

1) x + x = 6
1) ∠FAB ≅ ∠ABF 2) 2x + x = 6
2) ∠ABF ≅ ∠CBD 3) 3x + 2x = 6
3) CE ≅ EA 4) x + 2 x = 6
4) CF ≅ FA 3

290 A circle is represented by the equation 293 Point A is located at (4, −7). The point is reflected
x 2 + (y + 3) 2 = 13. What are the coordinates of the
in the x-axis. Its image is located at
1) (−4, 7)
center of the circle and the length of the radius?
1) (0, 3) and 13 2) (−4, −7)
3) (4, 7)
2) (0, 3) and 13
4) (7, −4)
3) (0, −3) and 13
4) (0, −3) and 13

294 In ABC , m∠A = 95, m∠B = 50, and m∠C = 35.

Which expression correctly relates the lengths of
291 What is an equation of the line that passes through the sides of this triangle?
the point (−2, 5) and is perpendicular to the line 1) AB < BC < CA
2) AB < AC < BC
whose equation is y = 1 x + 5?
2 3) AC < BC < AB
1) y = 2x + 1 4) BC < AC < AB
2) y = −2x + 1
3) y = 2x + 9
4) y = −2x − 9

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295 What is the negation of the statement “Squares are 299 Which expression best describes the transformation
parallelograms”? shown in the diagram below?
1) Parallelograms are squares.
2) Parallelograms are not squares.
3) It is not the case that squares are
parallelograms.
4) It is not the case that parallelograms are
squares.

296 Line segment AB is tangent to circle O at A. Which

type of triangle is always formed when points A, B,
and O are connected?
1) right
2) obtuse
3) scalene
4) isosceles

1) same orientation; reflection

2) opposite orientation; reflection
297 One step in a construction uses the endpoints of AB 3) same orientation; translation
to create arcs with the same radii. The arcs 4) opposite orientation; translation
intersect above and below the segment. What is
the relationship of AB and the line connecting the
points of intersection of these arcs?
←→
1) collinear 300 In the diagram below of ACT , BE  AT .
2) congruent
3) parallel
4) perpendicular

298 If ABC ∼ ZXY , m∠A = 50, and m∠C = 30, what

is m∠X ?
1) 30
2) 50 If CB = 3, CA = 10, and CE = 6, what is the length
3) 80 of ET ?
4) 100 1) 5
2) 14
3) 20
4) 26

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301 The equation of a circle is (x − 2) 2 + (y + 4) 2 = 4. 302 The pentagon in the diagram below is formed by
Which diagram is the graph of the circle? five rays.

1)

What is the degree measure of angle x?

1) 72
2) 96
3) 108
4) 112
2)

303 Triangle ABC has vertices A(1, 3) , B(0, 1) , and

C(4, 0). Under a translation, A ′, the image point of
A, is located at (4, 4). Under this same translation,
point C ′ is located at
1) (7, 1)
3) 2) (5, 3)
3) (3, 2)
4) (1, −1)

304 What is the negation of the statement “The Sun is

shining”?
1) It is cloudy.
4) 2) It is daytime.
3) It is not raining.
4) The Sun is not shining.

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305 Which equation represents the circle whose center

309 The lines represented by the equations y + 1 x = 4
is (−2, 3) and whose radius is 5? 2
1) (x − 2) 2 + (y + 3) 2 = 5 and 3x + 6y = 12 are
1) the same line
2) (x + 2) 2 + (y − 3) 2 = 5
2) parallel
3) (x + 2) 2 + (y − 3) 2 = 25 3) perpendicular
4) (x − 2) 2 + (y + 3) 2 = 25 4) neither parallel nor perpendicular

306 In a coordinate plane, how many points are both 5 310 The diameter of a circle has endpoints at (−2, 3)
units from the origin and 2 units from the x-axis? and (6, 3). What is an equation of the circle?
1) 1 1) (x − 2) 2 + (y − 3) 2 = 16
2) 2
2) (x − 2) 2 + (y − 3) 2 = 4
3) 3
4) 4 3) (x + 2) 2 + (y + 3) 2 = 16
4) (x + 2) 2 + (y + 3) 2 = 4

307 Which set of numbers represents the lengths of the

sides of a triangle? 311 Which equation represents a line perpendicular to
1) {5, 18, 13} the line whose equation is 2x + 3y = 12?
2) {6, 17, 22} 1) 6y = −4x + 12
3) {16, 24, 7} 2) 2y = 3x + 6
4) {26, 8, 15} 3) 2y = −3x + 6
4) 3y = −2x + 12

308 Isosceles trapezoid ABCD has diagonals AC and

BD . If AC = 5x + 13 and BD = 11x − 5 , what is the 312 If the endpoints of AB are A(−4, 5) and B(2, −5) ,
value of x?
what is the length of AB ?
1) 28
1) 2 34
2) 10 3 2) 2
4
3) 3 3) 61
4) 8
4) 1
2

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313 Square LMNO is shown in the diagram below. 315 In an equilateral triangle, what is the difference
between the sum of the exterior angles and the sum
of the interior angles?
1) 180°
2) 120°
3) 90°
4) 60°

316 In PQR, PQ = 8 , QR = 12, and RP = 13 . Which

statement about the angles of PQR must be true?
1) m∠Q > m∠P > m∠R
What are the coordinates of the midpoint of 2) m∠Q > m∠R > m∠P
diagonal LN ? 3) m∠R > m∠P > m∠Q
  4) m∠P > m∠R > m∠Q
1)  4 1 , −2 1 
 2 2
 
2)  −3 1 , 3 1 
 2 2 
 1 1  317 In the diagram below of circle O, chords AB and
3)  −2 , 3 
 2 2 CD intersect at E.
 1 1 
4)  −2 , 4 
 2 2

314 Lines j and k intersect at point P. Line m is drawn

so that it is perpendicular to lines j and k at point P.
Which statement is correct?
1) Lines j and k are in perpendicular planes.
2) Line m is in the same plane as lines j and k.
3) Line m is parallel to the plane containing lines j
and k. If CE = 10, ED = 6 , and AE = 4 , what is the length
4) Line m is perpendicular to the plane containing of EB ?
lines j and k. 1) 15
2) 12
3) 6.7
4) 2.4

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318 In the diagram below of circle O, chords AD and 320 Given: y = 1 x − 3

4
BC intersect at E.
y = x 2 + 8x + 12
In which quadrant will the graphs of the given
equations intersect?
1) I
2) II
3) III
4) IV

321 The diagram below shows AB and DE .

Which relationship must be true?
1) CAE ≅ DBE
2) AEC ∼ BED
3) ∠ACB ≅ ∠CBD
4) CA ≅ DB

319 Which geometric principle is used to justify the

construction below?

Which transformation will move AB onto DE such

that point D is the image of point A and point E is
the image of point B?
1) A line perpendicular to one of two parallel 1) T 3, −3
lines is perpendicular to the other. 2) D1
2
2) Two lines are perpendicular if they intersect to
form congruent adjacent angles. 3) R 90°
3) When two lines are intersected by a transversal 4) ry= x
and alternate interior angles are congruent, the
lines are parallel.
4) When two lines are intersected by a transversal
and the corresponding angles are congruent, the
lines are parallel.

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322 If two different lines are perpendicular to the same 325 In the diagram below, under which transformation
plane, they are will A ′B ′C ′ be the image of ABC ?
1) collinear
2) coplanar
3) congruent
4) consecutive

323 The rectangle ABCD shown in the diagram below

will be reflected across the x-axis.

1) rotation
2) dilation
3) translation
4) glide reflection

326 How many common tangent lines can be drawn to

the two externally tangent circles shown below?

What will not be preserved?

1) slope of AB
2) parallelism of AB and CD
3) length of AB
4) measure of ∠A 1) 1
2) 2
3) 3
4) 4
324 What is an equation of the line that contains the
point (3, −1) and is perpendicular to the line whose
equation is y = −3x + 2?
1) y = −3x + 8 327 If the surface area of a sphere is represented by
2) y = −3x 144π , what is the volume in terms of π ?
1) 36π
3) y = 1 x 2) 48π
3
3) 216π
4) y = 1 x − 2 4) 288π
3

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328 A right circular cylinder has a volume of 1,000 331 In the diagram below, circle A and circle B are
cubic inches and a height of 8 inches. What is the shown.
radius of the cylinder to the nearest tenth of an
inch?
1) 6.3
2) 11.2
3) 19.8
4) 39.8

329 In plane P, lines m and n intersect at point A. If

line k is perpendicular to line m and line n at point
A, then line k is
1) contained in plane P
2) parallel to plane P What is the total number of lines of tangency that
are common to circle A and circle B?
3) perpendicular to plane P 1) 1
4) skew to plane P 2) 2
3) 3
4) 4

330 In the diagram below of circle O, chords AE and

DC intersect at point B, such that mAC = 36 and
332 What is the slope of a line perpendicular to the line
mDE = 20.
whose equation is y = − 2 x − 5?
3
1) − 3
2
2) − 2
3
3) 2
3
4) 3
2

What is m∠ABC ?
1) 56
2) 36
3) 28
4) 8

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333 Towns A and B are 16 miles apart. How many 335 What is the equation of a line that is parallel to the
points are 10 miles from town A and 12 miles from line whose equation is y = x + 2?
town B? 1) x + y = 5
1) 1 2) 2x + y = −2
2) 2 3) y − x = −1
3) 3
4) y − 2x = 3
4) 0

334 Lines k 1 and k 2 intersect at point E. Line m is 336 In triangles ABC and DEF, AB = 4 , AC = 5 ,
DE = 8 , DF = 10 , and ∠A ≅ ∠D. Which method
perpendicular to lines k 1 and k 2 at point E. could be used to prove ABC ∼ DEF ?
1) AA
2) SAS
3) SSS
4) ASA

337 In the diagram below of ABC , D is a point on AB ,

AC = 7 , AD = 6 , and BC = 18 .

Which statement is always true?

1) Lines k 1 and k 2 are perpendicular.
2) Line m is parallel to the plane determined by
lines k 1 and k 2 .
3) Line m is perpendicular to the plane
determined by lines k 1 and k 2 .
The length of DB could be
4) Line m is coplanar with lines k 1 and k 2 .
1) 5
2) 12
3) 19
4) 25

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338 In the diagram below, a right circular cone has a 341 In the diagram below, PS is a tangent to circle O at
diameter of 8 inches and a height of 12 inches.
point S, PQR is a secant, PS = x , PQ = 3 , and
PR = x + 18 .

What is the volume of the cone to the nearest cubic

inch?
1) 201
2) 481
3) 603
4) 804 What is the length of PS ?
1) 6
2) 9
3) 3
4) 27
339 Secants JKL and JMN are drawn to circle O from
an external point, J. If JK = 8 , LK = 4 , and JM = 6 ,
what is the length of JN ?
1) 16
2) 12 342 Point A is not contained in plane B. How many
3) 10 lines can be drawn through point A that will be
4) 8 perpendicular to plane B?
1) one
2) two
3) zero
340 What is the solution of the following system of 4) infinite
equations?
y = (x + 3) 2 − 4
y = 2x + 5
1) (0, −4)
2) (−4, 0)
3) (−4, −3) and (0, 5)
4) (−3, −4) and (5, 0)

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343 Which diagram shows the construction of an 344 A rectangular prism has a volume of
equilateral triangle? 3x 2 + 18x + 24. Its base has a length of x + 2 and a
width of 3. Which expression represents the height
of the prism?
1) x + 4
2) x + 2
3) 3
4) x 2 + 6x + 8

1)

345 A right circular cylinder has an altitude of 11 feet

and a radius of 5 feet. What is the lateral area, in
square feet, of the cylinder, to the nearest tenth?
1) 172.7
2) 172.8
3) 345.4
4) 345.6
2)

346 In the diagram of trapezoid ABCD below, diagonals

AC and BD intersect at E and ABC ≅ DCB.

3)

Which statement is true based on the given

information?
1) AC ≅ BC
2) CD ≅ AD
4) 3) ∠CDE ≅ ∠BAD
4) ∠CDB ≅ ∠BAC

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347 In the diagram of ABC and DEF below, 349 What is an equation of a circle with its center at
AB ≅ DE , ∠A ≅ ∠D, and ∠B ≅ ∠E . (−3, 5) and a radius of 4?
1) (x − 3) 2 + (y + 5) 2 = 16
2) (x + 3) 2 + (y − 5) 2 = 16
3) (x − 3) 2 + (y + 5) 2 = 4
4) (x + 3) 2 + (y − 5) 2 = 4

350 What is the slope of a line perpendicular to the line

Which method can be used to prove whose equation is 2y = −6x + 8?
ABC ≅ DEF ? 1) −3
1) SSS
2) SAS 2) 1
6
3) ASA
4) HL 3) 1
3
4) −6

348 Which equation represents circle K shown in the

graph below? 351 What is an equation for the circle shown in the
graph below?

1) (x + 5) 2 + (y − 1) 2 = 3
2) (x + 5) 2 + (y − 1) 2 = 9
2 2
1) x2 + y2 = 2
3) (x − 5) + (y + 1) = 3
2) x2 + y2 = 4
4) (x − 5) 2 + (y + 1) 2 = 9
3) x2 + y2 = 8
4) x 2 + y 2 = 16

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352 On the set of axes below, Geoff drew rectangle 354 In isosceles trapezoid ABCD, AB ≅ CD. If
ABCD. He will transform the rectangle by using BC = 20 , AD = 36 , and AB = 17 , what is the length
the translation (x, y) → (x + 2, y + 1) and then will of the altitude of the trapezoid?
reflect the translated rectangle over the x-axis. 1) 10
2) 12
3) 15
4) 16

355 In the diagram below of circle C, mQT = 140, and

m∠P = 40.

What will be the area of the rectangle after these

transformations?
1) exactly 28 square units
2) less than 28 square units
3) greater than 28 square units
4) It cannot be determined from the information
What is mRS ?
given.
1) 50
2) 60
3) 90
4) 110
353 The equation of a circle is x 2 + (y − 7) 2 = 16. What
are the center and radius of the circle?
1) center = (0, 7); radius = 4
2) center = (0, 7); radius = 16 356 Which equation represents a line parallel to the line
3) center = (0, −7); radius = 4 whose equation is 2y − 5x = 10?
4) center = (0, −7); radius = 16 1) 5y − 2x = 25
2) 5y + 2x = 10
3) 4y − 10x = 12
4) 2y + 10x = 8

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357 The endpoints of CD are C(−2, −4) and D(6, 2) . 360 In the diagram below, ABC is inscribed in circle
P. The distances from the center of circle P to each
What are the coordinates of the midpoint of CD?
side of the triangle are shown.
1) (2, 3)
2) (2, −1)
3) (4, −2)
4) (4, 3)

358 The vertices of ABC are A(−1, −2), B(−1, 2) and

C(6, 0). Which conclusion can be made about the
angles of ABC ?
1) m∠A = m∠B
2) m∠A = m∠C
3) m∠ACB = 90
4) m∠ABC = 60 Which statement about the sides of the triangle is
true?
1) AB > AC > BC
2) AB < AC and AC > BC
359 In the diagram below, line k is perpendicular to 3) AC > AB > BC
plane P at point T. 4) AC = AB and AB > BC

361 A transformation of a polygon that always

preserves both length and orientation is
1) dilation
2) translation
3) line reflection
4) glide reflection

Which statement is true?

1) Any point in plane P also will be on line k.
2) Only one line in plane P will intersect line k.
3) All planes that intersect plane P will pass
through T.
4) Any plane containing line k is perpendicular to
plane P.

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362 In the diagram below, tangent AB and secant ACD 365 The diagram below shows the construction of the
are drawn to circle O from an external point A, center of the circle circumscribed about ABC .
AB = 8 , and AC = 4 .

What is the length of CD?

1) 16
2) 13
3) 12 This construction represents how to find the
4) 10 intersection of
1) the angle bisectors of ABC
2) the medians to the sides of ABC
3) the altitudes to the sides of ABC
363 What is the contrapositive of the statement, “If I am 4) the perpendicular bisectors of the sides of
ABC
tall, then I will bump my head”?
1) If I bump my head, then I am tall.
2) If I do not bump my head, then I am tall.
3) If I am tall, then I will not bump my head.
4) If I do not bump my head, then I am not tall. 366 Two triangles are similar, and the ratio of each pair
of corresponding sides is 2 :1. Which statement
regarding the two triangles is not true?
1) Their areas have a ratio of 4 :1.
2) Their altitudes have a ratio of 2 :1.
364 If the diagonals of a quadrilateral do not bisect
3) Their perimeters have a ratio of 2 :1.
each other, then the quadrilateral could be a
4) Their corresponding angles have a ratio of 2 :1.
1) rectangle
2) rhombus
3) square
4) trapezoid

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367 In the diagram below of trapezoid RSUT, RS  TU , 370 Which expression represents the volume, in cubic
centimeters, of the cylinder represented in the
X is the midpoint of RT , and V is the midpoint of diagram below?
SU .

If RS = 30 and XV = 44 , what is the length of TU ?

1) 37
2) 58
3) 74
4) 118
1) 162π
2) 324π
3) 972π
4) 3, 888π
368 The coordinates of the vertices of parallelogram
ABCD are A(−3, 2) , B(−2, −1) , C(4, 1), and D(3, 4) .
The slopes of which line segments could be
calculated to show that ABCD is a rectangle?
1) AB and DC 371 In the diagram of ABC below, AB ≅ AC . The
2) AB and BC measure of ∠B is 40°.
3) AD and BC
4) AC and BD

369 What is the slope of a line perpendicular to the line

whose equation is y = 3x + 4?
1) 1
3
2) − 1
3
What is the measure of ∠A?
3) 3
1) 40°
4) −3
2) 50°
3) 70°
4) 100°

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372 Side PQ of PQR is extended through Q to point 375 Tangents PA and PB are drawn to circle O from an
T. Which statement is not always true? external point, P, and radii OA and OB are drawn.
1) m∠RQT > m∠R If m∠APB = 40, what is the measure of ∠AOB?
2) m∠RQT > m∠P 1) 140º
3) m∠RQT = m∠P + m∠R 2) 100º
4) m∠RQT > m∠PQR 3) 70º
4) 50º

373 In the diagram below, the vertices of DEF are the

midpoints of the sides of equilateral triangle ABC, 376 The diagram below shows the construction of the
and the perimeter of ABC is 36 cm. bisector of ∠ABC .

What is the length, in centimeters, of EF ? Which statement is not true?

1) 6 1) m∠EBF = 1 m∠ABC
2) 12 2
3) 18 2) m∠DBF = 1 m∠ABC
4) 4 2
3) m∠EBF = m∠ABC
4) m∠DBF = m∠EBF

374 What is the equation of a line that passes through

the point (−3, −11) and is parallel to the line whose
equation is 2x − y = 4? 377 What is an equation of the line that passes through
the point (7, 3) and is parallel to the line
1) y = 2x + 5
4x + 2y = 10?
2) y = 2x − 5
1) y = 1 x − 1
3) y = 1 x + 25 2 2
2 2
2) y = − x + 13
1
4) y = − 1 x − 25 2 2
2 2
3) y = 2x − 11
4) y = −2x + 17

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378 In the diagram below, ABC is shown with AC 381 The diagonal AC is drawn in parallelogram ABCD.
extended through point D. Which method can not be used to prove that
ABC ≅ CDA?
1) SSS
2) SAS
3) SSA
4) ASA

382 In the diagram below, SQ and PR intersect at T, PQ

If m∠BCD = 6x + 2, m∠BAC = 3x + 15, and is drawn, and PS  QR.
m∠ABC = 2x − 1, what is the value of x?
1) 12
2) 14 10
11
3) 16
4) 18 1
9

379 A polygon is transformed according to the rule:

(x, y) → (x + 2, y). Every point of the polygon
moves two units in which direction?
1) up
2) down
3) left
4) right

What technique can be used to prove that

PST ∼ RQT ?
1) SAS
380 Two lines are represented by the equations
2) SSS
1
− y = 6x + 10 and y = mx. For which value of m 3) ASA
2
will the lines be parallel? 4) AA
1) −12
2) −3
3) 3
4) 12

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383 The diagram below shows the construction of a line 385 In the diagram below of ADB, m∠BDA = 90,
through point P perpendicular to line m. AD = 5 2 , and AB = 2 15 .

What is the length of BD ?

1) 10
2) 20
Which statement is demonstrated by this 3) 50
construction? 4) 110
1) If a line is parallel to a line that is
perpendicular to a third line, then the line is
also perpendicular to the third line.
2) The set of points equidistant from the
endpoints of a line segment is the 386 The diagram below illustrates the construction of
←→ ←
→
perpendicular bisector of the segment. PS parallel to RQ through point P.
3) Two lines are perpendicular if they are
equidistant from a given point.
4) Two lines are perpendicular if they intersect to
form a vertical line.

384 Which transformation is not always an isometry?

1) rotation
2) dilation
3) reflection
4) translation
Which statement justifies this construction?
1) m∠1 = m∠2
2) m∠1 = m∠3
3) PR ≅ RQ
4) PS ≅ RQ

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387 In which triangle do the three altitudes intersect 391 In the diagram of circle O below, chord CD is
outside the triangle?
parallel to diameter AOB and mAC = 30.
1) a right triangle
2) an acute triangle
3) an obtuse triangle
4) an equilateral triangle

388 Which transformation produces a figure similar but

not congruent to the original figure?
1) T 1, 3
2) D1
2 What is mCD?
3) R 90° 1) 150
4) ry= x 2) 120
3) 100
4) 60

389 Which transformation can map the letter S onto

itself? 392 In the diagram below, tangent PA and secant PBC
1) glide reflection are drawn to circle O from external point P.
2) translation
3) line reflection
4) rotation

390 In isosceles triangle ABC, AB = BC . Which

statement will always be true?
1) m∠B = m∠A
2) m∠A > m∠B
3) m∠A = m∠C
If PB = 4 and BC = 5 , what is the length of PA ?
4) m∠C < m∠B
1) 20
2) 9
3) 8
4) 6

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393 Which graph represents a circle with the equation 394 The figure in the diagram below is a triangular
(x − 5) 2 + (y + 1) 2 = 9? prism.

1)

Which statement must be true?

1) DE ≅ AB
2) AD ≅ BC
3) AD  CE
4) DE  BC

2)

395 In the diagram below of ACT , D is the midpoint

of AC , O is the midpoint of AT , and G is the
midpoint of CT .

3)

If AC = 10 , AT = 18 , and CT = 22, what is the

perimeter of parallelogram CDOG?
1) 21
2) 25
4) 3) 32
4) 40

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396 What is the converse of the statement "If Bob does 399 Given the equations: y = x 2 − 6x + 10
his homework, then George gets candy"?
1) If George gets candy, then Bob does his y+x=4
homework. What is the solution to the given system of
2) Bob does his homework if and only if George equations?
gets candy. 1) (2, 3)
3) If George does not get candy, then Bob does 2) (3, 2)
not do his homework. 3) (2, 2) and (1,3)
4) If Bob does not do his homework, then George 4) (2, 2) and (3, 1)
does not get candy.

400 What is the length of the line segment with

397 Based on the diagram below, which statement is endpoints (−6, 4) and (2, −5)?
true?
1) 13
2) 17
3) 72
4) 145

AB 3
401 Given ABC ∼ DEF such that = . Which
DE 2
statement is not true?
1) BC = 3
EF 2
1) ab m∠A 3
2) =
2) ac m∠D 2
3) bc area of ABC 9
3) =
4) de area of DEF 4
perimeter of ABC 3
4) =
perimeter of DEF 2

398 Juliann plans on drawing ABC , where the

measure of ∠A can range from 50° to 60° and the
measure of ∠B can range from 90° to 100°. Given 402 In which polygon does the sum of the measures of
these conditions, what is the correct range of the interior angles equal the sum of the measures of
measures possible for ∠C ? the exterior angles?
1) 20° to 40° 1) triangle
2) 30° to 50° 2) hexagon
3) 80° to 90° 3) octagon
4) 120° to 130° 4) quadrilateral

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403 In the diagram below of circle O, chords AD and 406 In the diagram below of circle O,
BC intersect at E, mAC = 87, and mBD = 35. chord AB  chord CD, and chord CD  chord EF .

Which statement must be true?

What is the degree measure of ∠CEA? 1) CE ≅ DF
1) 87 2) AC ≅ DF
2) 61
3) AC ≅ CE
3) 43.5
4) 26 4) EF ≅ CD

404 A support beam between the floor and ceiling of a 407 In the diagram below of AGE and OLD,
house forms a 90º angle with the floor. The builder ∠GAE ≅ ∠LOD, and AE ≅ OD.
wants to make sure that the floor and ceiling are
parallel. Which angle should the support beam
form with the ceiling?
1) 45º
2) 60º
3) 90º
4) 180º

To prove that AGE and OLD are congruent by

SAS, what other information is needed?
405 What are the center and radius of a circle whose
1) GE ≅ LD
equation is (x − A) 2 + (y − B) 2 = C ?
2) AG ≅ OL
1) center = (A, B); radius = C 3) ∠AGE ≅ ∠OLD
2) center = (−A, −B); radius = C 4) ∠AEG ≅ ∠ODL
3) center = (A, B); radius = C
4) center = (−A, −B); radius = C

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408 What are the center and the radius of the circle 411 In the diagram below of ABC , CD is the bisector
whose equation is (x − 3) 2 + (y + 3) 2 = 36 of ∠BCA, AE is the bisector of ∠CAB, and BG is
1) center = (3, −3); radius = 6 drawn.
2) center = (−3, 3); radius = 6
3) center = (3, −3); radius = 36
4) center = (−3, 3); radius = 36

409 Based on the construction below, which statement

must be true?

Which statement must be true?

1) DG = EG
2) AG = BG
3) ∠AEB ≅ ∠AEC
4) ∠DBG ≅ ∠EBG

412 What is the distance between the points (−3, 2) and

(1, 0)?
1) 2 2
2) 2 3
1 3) 5 2
1) m∠ABD = m∠CBD
2
4) 2 5
2) m∠ABD = m∠CBD
3) m∠ABD = m∠ABC
1
4) m∠CBD = m∠ABD
2
413 In three-dimensional space, two planes are parallel
and a third plane intersects both of the parallel
planes. The intersection of the planes is a
1) plane
410 If a line segment has endpoints A(3x + 5, 3y) and
2) point
B(x − 1, −y) , what are the coordinates of the 3) pair of parallel lines
midpoint of AB ? 4) pair of intersecting lines
1) (x + 3, 2y)
2) (2x + 2, y)
3) (2x + 3, y)
4) (4x + 4, 2y)

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414 In the diagram below of parallelogram STUV, 417 In the diagram of circle O below, chords AB and
SV = x + 3, VU = 2x − 1, and TU = 4x − 3.
CD are parallel, and BD is a diameter of the circle.

If mAD = 60, what is m∠CDB?

1) 20
2) 30
3) 60
4) 120

What is the length of SV ?

1) 5
2) 2 418 In the diagram below of circle O, secant AB
3) 7 intersects circle O at D, secant AOC intersects
4) 4 circle O at E, AE = 4 , AB = 12 , and DB = 6 .

415 Line k is drawn so that it is perpendicular to two

distinct planes, P and R. What must be true about
planes P and R?
1) Planes P and R are skew.
2) Planes P and R are parallel.
3) Planes P and R are perpendicular.
4) Plane P intersects plane R but is not
perpendicular to plane R.

What is the length of OC ?

1) 4.5
416 The lines 3y + 1 = 6x + 4 and 2y + 1 = x − 9 are 2) 7
3) 9
1) parallel
2) perpendicular 4) 14
3) the same line
4) neither parallel nor perpendicular

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419 Through a given point, P, on a plane, how many 422 In the diagram of circle O below, chord AB
lines can be drawn that are perpendicular to that
intersects chord CD at E, DE = 2x + 8 , EC = 3 ,
plane?
AE = 4x − 3 , and EB = 4 .
1) 1
2) 2
3) more than 2
4) none

420 Which statement is logically equivalent to "If it is

warm, then I go swimming"
1) If I go swimming, then it is warm.
2) If it is warm, then I do not go swimming.
3) If I do not go swimming, then it is not warm.
What is the value of x?
4) If it is not warm, then I do not go swimming.
1) 1
2) 3.6
3) 5
4) 10.25
421 The diagram below shows a right pentagonal prism.

423 In the diagram below, which transformation was

used to map ABC to A ′B ′C ′?

1) dilation
Which statement is always true? 2) rotation
3) reflection
1) BC  ED 4) glide reflection
2) FG  CD
3) FJ  IH
4) GB  HC

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424 Which geometric principle is used in the 427 The diagram below shows the construction of the
construction shown below? perpendicular bisector of AB.

1) The intersection of the angle bisectors of a

triangle is the center of the inscribed circle.
2) The intersection of the angle bisectors of a
triangle is the center of the circumscribed
circle.
3) The intersection of the perpendicular bisectors Which statement is not true?
of the sides of a triangle is the center of the 1) AC = CB
inscribed circle. 2) CB = 1 AB
4) The intersection of the perpendicular bisectors 2
of the sides of a triangle is the center of the 3) AC = 2AB
circumscribed circle. 4) AC + CB = AB

425 What is the perimeter of a rhombus whose 428 In ABC , AB = 7 , BC = 8 , and AC = 9 . Which list
diagonals are 16 and 30? has the angles of ABC in order from smallest to
1) 92 largest?
2) 68 1) ∠A, ∠B, ∠C
3) 60 2) ∠B, ∠A, ∠C
4) 17 3) ∠C, ∠B, ∠A
4) ∠C, ∠A, ∠B

426 What is the measure of an interior angle of a

regular octagon?
1) 45º
2) 60º
3) 120º
4) 135º

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Geometry Multiple Choice Regents Exam Questions
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429 In the diagram below, circle O has a radius of 5, 432 What is the negation of the statement “I am not
and CE = 2. Diameter AC is perpendicular to going to eat ice cream”?
1) I like ice cream.
chord BD at E.
2) I am going to eat ice cream.
3) If I eat ice cream, then I like ice cream.
4) If I don’t like ice cream, then I don’t eat ice
cream.

433 ABC is similar to DEF . The ratio of the length

of AB to the length of DE is 3 :1. Which ratio is
also equal to 3 :1?
1) m∠A
m∠D
What is the length of BD ?
1) 12 2) m∠B
m∠F
2) 10 area of ABC
3) 8 3)
area of DEF
4) 4 perimeter of ABC
4)
perimeter of DEF

430 The endpoints of AB are A(3, 2) and B(7, 1) . If

A ″B ″ is the result of the transformation of AB 434 What is the slope of a line perpendicular to the line
under D 2  T −4, 3 what are the coordinates of A ″ and whose equation is 5x + 3y = 8?
B ″? 1) 5
1) A ″(−2, 10) and B ″(6, 8) 3
2) A ″(−1, 5) and B ″(3, 4) 2) 3
5
3) A ″(2, 7) and B ″(10, 5)
4) A ″(14, −2) and B ″(22, −4) 3) − 3
5
4) − 5
3

431 What is the image of point A(4, 2) after the

composition of transformations defined by
R 90°  r y = x ?
1) (−4, 2)
2) (4, −2)
3) (−4, −2)
4) (2, −4)

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Geometry Multiple Choice Regents Exam Questions
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435 In the diagram below, ABC ≅ XYZ . 437 Which graph could be used to find the solution to
the following system of equations?
y = −x + 2
y = x2

Which two statements identify corresponding

congruent parts for these triangles?
1) AB ≅ XY and ∠C ≅ ∠Y
2) AB ≅ YZ and ∠C ≅ ∠X
1)
3) BC ≅ XY and ∠A ≅ ∠Y
4) BC ≅ YZ and ∠A ≅ ∠X

436 In the diagram below of PRT , Q is a point on PR ,

S is a point on TR, QS is drawn, and
∠RPT ≅ ∠RSQ.
2)

3)

Which reason justifies the conclusion that

PRT ∼ SRQ?
1) AA
2) ASA
3) SAS
4) SSS
4)

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Geometry Multiple Choice Regents Exam Questions
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438 What is the slope of a line that is perpendicular to 441 After a composition of transformations, the
the line whose equation is 3x + 4y = 12? coordinates A(4, 2) , B(4, 6) , and C(2, 6) become
A ″(−2, −1), B ″(−2, −3), and C ″(−1, −3), as shown on
1) 3
4 the set of axes below.
2) − 3
4
3) 4
3
4) − 4
3

439 What is the length, to the nearest tenth, of the line

segment joining the points (−4, 2) and (146, 52)?
1) 141.4
2) 150.5
3) 151.9
4) 158.1
Which composition of transformations was used?
1) R 180°  D 2
2) R 90°  D 2
440 In the diagram below of right triangle ACB, altitude 3) D 1  R 180°
2
CD is drawn to hypotenuse AB .
4) D 1  R 90°
2

442 In ABC , AB ≅ BC . An altitude is drawn from B

to AC and intersects AC at D. Which conclusion is
not always true?
1) ∠ABD ≅ ∠CBD
If AB = 36 and AC = 12 , what is the length of AD ? 2) ∠BDA ≅ ∠BDC
1) 32 3) AD ≅ BD
2) 6
4) AD ≅ DC
3) 3
4) 4

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Geometry Multiple Choice Regents Exam Questions
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443 What is the inverse of the statement “If two 446 Given the system of equations: y = x 2 − 4x
triangles are not similar, their corresponding angles
are not congruent”? x=4
1) If two triangles are similar, their corresponding The number of points of intersection is
angles are not congruent. 1) 1
2) If corresponding angles of two triangles are not 2) 2
congruent, the triangles are not similar. 3) 3
3) If two triangles are similar, their corresponding 4) 0
angles are congruent.
4) If corresponding angles of two triangles are
congruent, the triangles are similar.
447 In the diagram of ABC and EDC below, AE
and BD intersect at C, and ∠CAB ≅ ∠CED.

444 Which illustration shows the correct construction

of an angle bisector?

1)

Which method can be used to show that ABC

must be similar to EDC ?
2)
1) SAS
2) AA
3) SSS
4) HL
3)

4) 448 In ABC , point D is on AB , and point E is on BC

such that DE  AC . If DB = 2 , DA = 7 , and
DE = 3 , what is the length of AC ?
1) 8
445 Which transformation of the line x = 3 results in an
2) 9
image that is perpendicular to the given line?
3) 10.5
1) r x-axis
4) 13.5
2) r y-axis
3) ry= x
4) rx = 1

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