The University of the State of New York
REGENTS HIGH SCHOOL EXAMINATION
GEOMETRY
(
COMMON CORE
)
Friday, June 17, 2016 — 1:15 to 4:15 p.m.
MODEL RESPONSE SET
Table of Contents
Question 25 . . . . . . . . . . . . . . . . . . . 2
Question 26 . . . . . . . . . . . . . . . . . . 11
Question 27 . . . . . . . . . . . . . . . . . . 17
Question 28 . . . . . . . . . . . . . . . . . . 23
Question 29 . . . . . . . . . . . . . . . . . . 27
Question 30 . . . . . . . . . . . . . . . . . . 33
Question 31 . . . . . . . . . . . . . . . . . . 38
Question 32 . . . . . . . . . . . . . . . . . . 43
Question 33 . . . . . . . . . . . . . . . . . . 50
Question 34 . . . . . . . . . . . . . . . . . . 56
Question 35 . . . . . . . . . . . . . . . . . . 64
Question 36 . . . . . . . . . . . . . . . . . . 74
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [2]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [3]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [4]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [5]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 1: The student gave a correct description of the reflection, but gave an incomplete
description of the translation.
Geometry (Common Core) – June ’16 [6]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 1: The student gave a correct description of the reflection, but the description of the
rotation did not include the center.
Geometry (Common Core) – June ’16 [7]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 1: The student described an appropriate sequence, but the description was incomplete.
Geometry (Common Core) – June ’16 [8]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 1: The student graphed the transformation correctly, but did not write a description.
Geometry (Common Core) – June ’16 [9]
Question 25
25 Describe a sequence of transformations that will map ABC onto DEF as shown below.
y
x
C
BA
F
E
D
Score 0: The student gave an incomplete description of the reflection (flip) and described the
translation (move) incorrectly.
Geometry (Common Core) – June ’16 [10]
Geometry (Common Core) – June ’16 [11]
Question 26
26 Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,2), and B has
coordinates (22,2), determine and state the coordinates of P.
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [12]
Question 26
26 Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,2), and B has
coordinates (22,2), determine and state the coordinates of P.
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [13]
Question 26
26 Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,2), and B has
coordinates (22,2), determine and state the coordinates of P.
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [14]
Question 26
26 Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,2), and B has
coordinates (22,2), determine and state the coordinates of P.
Score 1: The student showed correct work to partition the segment in a 5:4 ratio.
Geometry (Common Core) – June ’16 [15]
Question 26
26 Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,2), and B has
coordinates (22,2), determine and state the coordinates of P.
Score 1: The student showed correct work to determine the x-coordinate of P, but made an
error in determining the y-coordinate.
Geometry (Common Core) – June ’16 [16]
Question 26
26 Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,2), and B has
coordinates (22,2), determine and state the coordinates of P.
Score 0: The student determined the correct y-coordinate by calculating the midpoint of
–—–—
AB,
but the midpoint was not relevant to the problem.
Geometry (Common Core) – June ’16 [17]
Question 27
27 In CED as shown below, points A and B are located on sides
–—–—
CE and
–—–—
ED, respectively. Line
segment AB is drawn such that AE ⫽ 3.75, AC ⫽ 5, EB ⫽ 4.5, and BD ⫽ 6.
Explain why
–—–—
AB is parallel to
–—–—
CD.
5
3.75
A
C
D
E
4.5
6
B
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [18]
Question 27
27 In CED as shown below, points A and B are located on sides
–—–—
CE and
–—–—
ED, respectively. Line
segment AB is drawn such that AE ⫽ 3.75, AC ⫽ 5, EB ⫽ 4.5, and BD ⫽ 6.
Explain why
–—–—
AB is parallel to
–—–—
CD.
5
3.75
A
C
D
E
4.5
6
B
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [19]
Question 27
27 In CED as shown below, points A and B are located on sides
–—–—
CE and
–—–—
ED, respectively. Line
segment AB is drawn such that AE ⫽ 3.75, AC ⫽ 5, EB ⫽ 4.5, and BD ⫽ 6.
Explain why
–—–—
AB is parallel to
–—–—
CD.
5
3.75
A
C
D
E
4.5
6
B
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [20]
Question 27
27 In CED as shown below, points A and B are located on sides
–—–—
CE and
–—–—
ED, respectively. Line
segment AB is drawn such that AE ⫽ 3.75, AC ⫽ 5, EB ⫽ 4.5, and BD ⫽ 6.
Explain why
–—–—
AB is parallel to
–—–—
CD.
5
3.75
A
C
D
E
4.5
6
B
Score 1: The student showed that the cross products of the proportion are equal, but the
explanation was incorrect.
Geometry (Common Core) – June ’16 [21]
Question 27
27 In CED as shown below, points A and B are located on sides
–—–—
CE and
–—–—
ED, respectively. Line
segment AB is drawn such that AE ⫽ 3.75, AC ⫽ 5, EB ⫽ 4.5, and BD ⫽ 6.
Explain why
–—–—
AB is parallel to
–—–—
CD.
5
3.75
A
C
D
E
4.5
6
B
Score 0: The student only wrote a correct proportion.
Geometry (Common Core) – June ’16 [22]
Question 27
27 In CED as shown below, points A and B are located on sides
–—–—
CE and
–—–—
ED, respectively. Line
segment AB is drawn such that AE ⫽ 3.75, AC ⫽ 5, EB ⫽ 4.5, and BD ⫽ 6.
Explain why
–—–—
AB is parallel to
–—–—
CD.
5
3.75
A
C
D
E
4.5
6
B
Score 0: The student had a completely incorrect response.
Geometry (Common Core) – June ’16 [23]
Question 28
Score 2: The student had a complete and correct response.
28 Find the value of R that will make the equation sin 73° ⫽ cos R true when 0° ⬍ R ⬍ 90°.
Explain your answer.
Geometry (Common Core) – June ’16 [24]
Question 28
Score 1: The student correctly determined the value of R, but the explanation was missing.
28 Find the value of R that will make the equation sin 73° ⫽ cos R true when 0° ⬍ R ⬍ 90°.
Explain your answer.
Geometry (Common Core) – June ’16 [25]
Question 28
Score 1: The student correctly determined the value of R, but the explanation was incorrect.
28 Find the value of R that will make the equation sin 73° ⫽ cos R true when 0° ⬍ R ⬍ 90°.
Explain your answer.
Geometry (Common Core) – June ’16 [26]
Question 28
Score 0: The student had a completely incorrect response.
28 Find the value of R that will make the equation sin 73° ⫽ cos R true when 0° ⬍ R ⬍ 90°.
Explain your answer.
Geometry (Common Core) – June ’16 [27]
Question 29
29 In the diagram below, Circle 1 has radius 4, while Circle 2 has radius 6.5. Angle A intercepts an
arc of length π, and angle B intercepts an arc of length .
Dominic thinks that angles A and B have the same radian measure. State whether Dominic is
correct or not. Explain why.
13
8
π
Circle 2
6.5
B
13π
8
π
A
4
Circle 1
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [28]
Question 29
29 In the diagram below, Circle 1 has radius 4, while Circle 2 has radius 6.5. Angle A intercepts an
arc of length π, and angle B intercepts an arc of length .
Dominic thinks that angles A and B have the same radian measure. State whether Dominic is
correct or not. Explain why.
13
8
π
Circle 2
6.5
B
13π
8
π
A
4
Circle 1
Score 2: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [29]
Question 29
29 In the diagram below, Circle 1 has radius 4, while Circle 2 has radius 6.5. Angle A intercepts an
arc of length π, and angle B intercepts an arc of length .
Dominic thinks that angles A and B have the same radian measure. State whether Dominic is
correct or not. Explain why.
13
8
π
Circle 2
6.5
B
13π
8
π
A
4
Circle 1
Score 1: The student made an error in transcribing , but wrote a correct explanation based
on the error.
13
8
π
Geometry (Common Core) – June ’16 [30]
Question 29
29 In the diagram below, Circle 1 has radius 4, while Circle 2 has radius 6.5. Angle A intercepts an
arc of length π, and angle B intercepts an arc of length .
Dominic thinks that angles A and B have the same radian measure. State whether Dominic is
correct or not. Explain why.
13
8
π
Circle 2
6.5
B
13π
8
π
A
4
Circle 1
Score 1: The student wrote a correct proportion and showed work with a correct conclusion, but
the explanation was missing.
Geometry (Common Core) – June ’16 [31]
Question 29
29 In the diagram below, Circle 1 has radius 4, while Circle 2 has radius 6.5. Angle A intercepts an
arc of length π, and angle B intercepts an arc of length .
Dominic thinks that angles A and B have the same radian measure. State whether Dominic is
correct or not. Explain why.
13
8
π
Circle 2
6.5
B
13π
8
π
A
4
Circle 1
Score 0: The student wrote a correct proportion, but no explanation was written.
Geometry (Common Core) – June ’16 [32]
Question 29
29 In the diagram below, Circle 1 has radius 4, while Circle 2 has radius 6.5. Angle A intercepts an
arc of length π, and angle B intercepts an arc of length .
Dominic thinks that angles A and B have the same radian measure. State whether Dominic is
correct or not. Explain why.
13
8
π
Circle 2
6.5
B
13π
8
π
A
4
Circle 1
Score 0: The student had a completely incorrect response.
Geometry (Common Core) – June ’16 [33]
Question 30
Score 2: The student had a complete and correct response.
30 A ladder leans against a building. The top of the ladder touches the building 10 feet above the
ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the angle
that the ladder makes with the level ground.
Geometry (Common Core) – June ’16 [34]
Question 30
Score 2: The student had a complete and correct response.
30 A ladder leans against a building. The top of the ladder touches the building 10 feet above the
ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the angle
that the ladder makes with the level ground.
Geometry (Common Core) – June ’16 [35]
Question 30
Score 1: The student wrote a correct trigonometric equation.
30 A ladder leans against a building. The top of the ladder touches the building 10 feet above the
ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the angle
that the ladder makes with the level ground.
Geometry (Common Core) – June ’16 [36]
Question 30
Score 1: The student incorrectly labeled the height, but found an appropriate angle measure.
30 A ladder leans against a building. The top of the ladder touches the building 10 feet above the
ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the angle
that the ladder makes with the level ground.
Geometry (Common Core) – June ’16 [37]
Question 30
Score 0: The student used the Pythagorean Theorem to find the length of the ladder and made
no attempt to find the measure of the angle.
30 A ladder leans against a building. The top of the ladder touches the building 10 feet above the
ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the angle
that the ladder makes with the level ground.
Geometry (Common Core) – June ’16 [38]
Question 31
Score 2: The student had a complete and correct response.
31 In the diagram below, radius
–—–—
OA is drawn in circle O. Using a compass and a straightedge,
construct a line tangent to circle O at point A. [Leave all construction marks.]
Geometry (Common Core) – June ’16 [39]
Question 31
Score 2: The student had a complete and correct response.
31 In the diagram below, radius
–—–—
OA is drawn in circle O. Using a compass and a straightedge,
construct a line tangent to circle O at point A. [Leave all construction marks.]
Geometry (Common Core) – June ’16 [40]
Question 31
Score 2: The student had a complete and correct response.
31 In the diagram below, radius
–—–—
OA is drawn in circle O. Using a compass and a straightedge,
construct a line tangent to circle O at point A. [Leave all construction marks.]
A
O
Geometry (Common Core) – June ’16 [41]
Question 31
Score 1: The student did not indicate the endpoint of the diameter of circle A, which was
necessary to construct the other arcs.
31 In the diagram below, radius
–—–—
OA is drawn in circle O. Using a compass and a straightedge,
construct a line tangent to circle O at point A. [Leave all construction marks.]
Geometry (Common Core) – June ’16 [42]
Question 31
Score 0: The student made a drawing that was not a construction.
31 In the diagram below, radius
–—–—
OA is drawn in circle O. Using a compass and a straightedge,
construct a line tangent to circle O at point A. [Leave all construction marks.]
Geometry (Common Core) – June ’16 [43]
Question 32
32 A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a
diameter of 22.5 inches and a height of 33.5 inches. There are 231 cubic inches in a liquid gallon.
Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil.
Score 4: The student had a complete and correct response.
Geometry (Common Core) – June ’16 [44]
Question 32
32 A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a
diameter of 22.5 inches and a height of 33.5 inches. There are 231 cubic inches in a liquid gallon.
Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil.
Score 3: The student made an error in calculating the volume.
Geometry (Common Core) – June ’16 [45]
Question 32
32 A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a
diameter of 22.5 inches and a height of 33.5 inches. There are 231 cubic inches in a liquid gallon.
Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil.
Score 2: The student did not multiply by π and made a rounding error.
Geometry (Common Core) – June ’16 [46]
Question 32
32 A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a
diameter of 22.5 inches and a height of 33.5 inches. There are 231 cubic inches in a liquid gallon.
Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil.
Score 1: The student made an error in using the diameter to find the volume of the barrel, and
did not find the number of gallons.
Geometry (Common Core) – June ’16 [47]
Question 32
32 A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a
diameter of 22.5 inches and a height of 33.5 inches. There are 231 cubic inches in a liquid gallon.
Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil.
Score 1: The student used an incorrect volume formula and made a rounding error.
Geometry (Common Core) – June ’16 [48]
Question 32
32 A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a
diameter of 22.5 inches and a height of 33.5 inches. There are 231 cubic inches in a liquid gallon.
Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil.
Score 1: The student made correct substitutions into the volume formula of a cylinder, but no
further correct work was shown.
Geometry (Common Core) – June ’16 [49]
Question 32
32 A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a
diameter of 22.5 inches and a height of 33.5 inches. There are 231 cubic inches in a liquid gallon.
Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil.
Score 0: The student used an incorrect formula, made a computational error, and did not
determine the number of gallons of fuel.
Geometry (Common Core) – June ’16 [50]
Question 33
Score 4: The student had a complete and correct proof.
33 Given: Parallelogram ABCD,
––—–—
EFG, and diagonal
––—–—
DFB
Prove: DEF ⬃ BGF
AB
G
CD
E
F
Geometry (Common Core) – June ’16 [51]
Question 33
Score 4: The student had a complete and correct response.
33 Given: Parallelogram ABCD,
––—–—
EFG, and diagonal
––—–—
DFB
Prove: DEF ⬃ BGF
AB
G
CD
E
F
Geometry (Common Core) – June ’16 [52]
Question 33
Score 3: The student omitted one statement and reason.
33 Given: Parallelogram ABCD,
––—–—
EFG, and diagonal
––—–—
DFB
Prove: DEF ⬃ BGF
AB
G
CD
E
F
Geometry (Common Core) – June ’16 [53]
Question 33
Score 2: The student made an error in assuming that
––—–—
DFB and
––—–—
EFG are both diagonals, which
significantly reduced the difficulty of the proof.
33 Given: Parallelogram ABCD,
––—–—
EFG, and diagonal
––—–—
DFB
Prove: DEF ⬃ BGF
AB
G
CD
E
F
Geometry (Common Core) – June ’16 [54]
Question 33
Score 1: The student had only one correct relevant statement and reason.
33 Given: Parallelogram ABCD,
––—–—
EFG, and diagonal
––—–—
DFB
Prove: DEF ⬃ BGF
AB
G
CD
E
F
Geometry (Common Core) – June ’16 [55]
Question 33
Score 0: The student had no correct reasons.
33 Given: Parallelogram ABCD,
––—–—
EFG, and diagonal
––—–—
DFB
Prove: DEF ⬃ BGF
AB
G
CD
E
F
Geometry (Common Core) – June ’16 [56]
Question 34
Score 4: The student had a complete and correct response.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
A⬘
B⬘
A
C⬘
B
C
y
x
Describe the transformation that was performed.
Geometry (Common Core) – June ’16 [57]
Question 34
Score 4: The student had a complete and correct response.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
A⬘
B⬘
A
C⬘
B
C
y
x
Describe the transformation that was performed.
Geometry (Common Core) – June ’16 [58]
Question 34
Score 3: The student did not state the center of dilation.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
Describe the transformation that was performed.
A⬘
B⬘
A
C⬘
B
C
y
x
Geometry (Common Core) – June ’16 [59]
Question 34
Score 2: The student did not state the center of dilation. The student explained why the angles
are congruent, but did not explain why the triangles are similar.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
A⬘
B⬘
A
C⬘
B
C
y
x
Describe the transformation that was performed.
Geometry (Common Core) – June ’16 [60]
Question 34
Score 1: The student did not state the scale factor of the dilation and did not write a correct
explanation.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
A⬘
B⬘
A
C⬘
B
C
y
x
Describe the transformation that was performed.
Geometry (Common Core) – June ’16 [61]
Question 34
Score 1: The student had an incomplete description of the dilation and an incorrect explanation
of the similar triangles.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
A⬘
B⬘
A
C⬘
B
C
y
x
Describe the transformation that was performed.
Geometry (Common Core) – June ’16 [62]
Question 34
S
core 0: The student did not describe the dilation, and had an incorrect explanation of the similar
t
riangles.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
A⬘
B⬘
A
C⬘
B
C
y
x
Describe the transformation that was performed.
Geometry (Common Core) – June ’16 [63]
Question 34
Score 0: The student had a completely incorrect response.
34 In the diagram below, A ⬘B⬘C⬘ is the image of ABC after a transformation.
Explain why A⬘B⬘C⬘ ⬃ ABC.
A⬘
B⬘
A
C⬘
B
C
y
x
Describe the transformation that was performed.
Geometry (Common Core) – June ’16 [64]
Question 35
Score 6: The student had a complete and correct proof.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [65]
Question 35
Score 6: The student had a complete and correct response.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [66]
Question 35
Score 5: The student had a statement and reason missing between steps 9 and 10.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [67]
Question 35
Score 4: The student had a statement and reason missing to prove step 3 and a statement and
reason missing to prove step 8.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [68]
Question 35
Score 4: The student had an incorrect reason in step 3 and an incomplete reason in step 4.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [69]
Question 35
Score 3: The student had an incorrect reason in proving the isosceles triangle, and no further
correct work was shown.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [70]
Question 35
Score 2: The student made one conceptual error in step 3 and had one missing statement and
reason to prove step 6.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [71]
Question 35
Score 2: The student used the incorrect parallel sides to conclude ∠1 ⬵ ∠3, had an incomplete
reason in step 4, and did not prove the right triangle.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [72]
Question 35
Score 1: The student had only two correct statements and reasons.
(Steps 2 and 4 can be combined.)
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [73]
Question 35
Score 0: The student had no correct work.
35 Given: Quadrilateral ABCD with diagonals
–—–—
AC and
–—–—
BD that bisect each other, and ∠1 ⬵ ∠2
Prove: ACD is an isosceles triangle and AEB is a right triangle
AD
B
C
1
2
E
Geometry (Common Core) – June ’16 [74]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [75]
Question 36
Score 6: The student had a complete and correct response.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
Geometry (Common Core) – June ’16 [76]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [77]
Question 36
Score 6: The student had a complete and correct response.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
Geometry (Common Core) – June ’16 [78]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [79]
Question 36
Score 5: The student made one rounding error.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
Geometry (Common Core) – June ’16 [80]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [81]
Question 36
Score 4: The student made a conceptual error in using the wrong formula in determine the volume
of the water glass.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
Geometry (Common Core) – June ’16 [82]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [83]
Question 36
Score 3: The student correctly determined the height and the volume of the larger cone.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
Geometry (Common Core) – June ’16 [84]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [85]
Question 36
Score 2: The student only found the correct value of the height.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
Geometry (Common Core) – June ’16 [86]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [87]
Question 36
Score 1: The student had a correct explanation.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
Geometry (Common Core) – June ’16 [88]
Question 36
36 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base)
as shown below.
The diameter of the top of the glass is 3 inches, the diameter at the bottom of the glass is 2 inches,
and the height of the glass is 5 inches.
The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in
order to find the height of the cone. Explain why.
Question 36 is continued on the next page.
Geometry (Common Core) – June ’16 [89]
Question 36
Score 0: The student had no correct work.
Question 36 continued
Determine and state, in inches, the height of the larger cone.
Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass.
