BA 13 X 0613geans
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GEOMETRY
The University of the State of New York
REGENTS HIGH SCHOOL EXAMINATION
GEOMETRY
Wednesday, June 19, 2013-9:15 a.m. to 12:15 p.m., only
Student Name:
c;; b()}
~~fl1--~~--;0---------------------------------
fY} (,
SchooiName: ____
The possession or use of any communications device is strictly prohibited when taking this
examination. If you have or use any communications device, no matter how briefly, your examination
will be invalidated and no score will be calculated for you.
Print your name and the name of your school on the lines above.
A separate answer sheet for Part I h~s been provided to you. Follow the
instructions from the proctor for completing the student information on your answer
sheet.
This examination has four parts, with a total of 38 questions. You must answer
all questions in this examination. Record your answers to the Part I multiple-choice
questions on the separate answer sheet. Write your answers to the questions in
Parts II, III, and IV directly in this booklet. All work should be written in pen,
except for graphs and drawings, which should be done in pencil. Clearly indicate
the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc.
The formulas that you may need to answer some questions in this examination
are found at the end of the examination. This sheet is perforated so you may
remove it from this booklet.
Scrap paper is not permitted for any part of this examination, but you may use
the blank spa~es in this booklet as scrap paper. A perforated sheet of scrap graph
paper is provided .at,,the end of this booklet for any question for which graphing
may be helpful-but is_Jlot required. You may remove this sheet from this booklet.
Any work done on this sheet of scrap graph paper will not be scored.
\
When yoti~ave completed the examination, you must sign the statement printed
at the end of the a'swer sheet, indicating that you had no unlawful knowledge of
the questi6'ns or ap:swers prior to the examination and that you have neither given
nor received assiltance in answering any of the questions during the examination.
Your answer sheet cannot be accepted if you fail to sign this declaration.
Notice ...
A graphing calculator, a straightedge (ruler), and a compass must be available for you to use while
taking this examination.
,,
DO NOT OPEN THIS EXAMINATION BOOKLET UNTIL TilE SIGNAL IS GIVEN.
Al::IJ.3V\103E>
Part I
Answer all 28 questions in this part. Each correct answer will receive 2 credits. For each
statement or question, choose the word or expression that, of those given, best completes the
statement or answers the question. Record your answers on your separate answer sheet. [56]
I In trapezoid RSTV below with bases RS and VT, diagonals RT and
SV intersect at Q.
R
~
S
If trapezoid RSTV is not isosceles, which triangle is equal in area to
6RSV?
(1) 6RQV
(3) 6RVT
@)6RST
(4) 6SVT
2 In the diagram below, 6XYV :::::: 6 TSV.
8
Which statement can not be proven?
(1) LXVY:::::: LTVS
(3) XY:::::: TS
@ LVYX:::::: LVUT
(4) YV:::::: SV
Geometry- June '13
[2]
Use this space for
computations.
Use this space for
computations.
3 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
(3) 3
(2) 2
@4
------·-.....L-
.-
-
•
~
\
-~ - ~ - - --
--
---------
---..,:..__
~
---"\,
\
~-
\
4 vVhat are the coordinates of A', the image of point A( -3,4), after a
rotation of 180° about the origin?
(l) (4, -3)
(2) ( -4, -3)
/
(3) (3,4)
()' J
y) c.-7
@(3, -4)
5 Based on the construction below, which conclusion is not always
tme?
(1) AB .1 CD
(3) AE = EB
@)AB =CD
(4) CE =DE
Geometry- June '13
[3]
[OVER]
6 Which equation represents the circle whose center is (- 5,.3) and
that passes through the point ( -1,.3)?
(I) (x
+
1) 2
(2) (x - 1)2
<[~
(x
+ 5) 2
(4) (x - 5) 2
+ (y - .3) 2 =
+ (y + .3) 2 =
+ (y - .3) 2 =
+ (y + .3) 2 =
16
16
16
16
7 As shown in the diagram below, when right triangle DAB is reflected
over the x-axis, its image is triangle DCB.
y
Which statement justifles why AB ::::: CB?
@Distance is preserved under reflection.
(2) Orientation is preserved under reflection.
(,3) Points on the line of reflection remain invariant.
(4) Right angles remain congruent under reflection.
Geometry - June '13
[4]
Use this space for
computations.
s~
~5
70
3(/c]r I
~(/g)-(1
r;(ig) )b
8 In .6.ABC, mLA = 3x + 1, mLB = 4x - 17, and mLC = .Sx - 20.
'3 X +I
\:Vhich type of triangle is D.ABC?
@ isos~eles
(1) right
(2) scalene
(4) eqmlateral
Use this space for
computations.
f· Cf) -·/7 -j-· S") - J 0 :: /8' 0
}J.-x _36 ~/ /86
/
. /{
b
d--.x -:; d
){:, !8
9 What is the equation for circle 0 shown in the graph below?
y
X
(1) (x - 3 )2 + (y + l )2
(2) (x
@ (x
(4) (x
=6
=6
+ 3) 2 + (y - 1) 2
- 3 )2 + (y + 1)2 = 9
+ 3) 2 + (y - 1) 2 = 9
Geometry- June '13
[5]
[OVER]
Use this space for
computations.
10 Point A is on line m. How many distinct planes will be perpendicular
to line m and pass through point A?
@one
(2) two
(3) zero
(4) infinite
11 In 6ABC, Dis the midpoint of AB and E is the midpoint of BC.
If AC = 3x - 15 and DE = 6, what is the value of x?
8
] X
-15~(J-.){b)
)x-)(;:/ J)_
3x > ),7
9
.Y~
(1) 6
(2) 7
@9
(4) 12
of its diameter are A(8, -4) and B( -3,2)?
(1) (2.5,1)
@(2.5, -1)
(3) (5.5, -3)
-~
)._.
?-
1
(4) (5.5,3)
{ £.
;)-
J
( .. · ).
-LJ
~
(),51 r/)
Geometry- June '13
J
(Qf!_!__!_,
._
12 What are the coordinates of the center of a circle if the endpoints
[6]
.
13 Which graph could be used to find the solution to the following
system of equations?
y = (x
+ 3)2 -
x+y=2
1
VtV1-f-'¥ ( ... ~I -
x rJ.-
Use this space for
computations.
d
y~)-y
X
(1)
(3)
y
y
X
X
(4)
14 What is the converse of .. If an angle measures 90 degrees, then it is
a right angle"?
~If an angle is a right angle, then it measures 90 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.
(4) If an angle does not measure 90 degrees, then it is not a right
angle.
Geometry -June '13
[7]
[OVER]
Use this space for
15 As shown in the diagram below, a right pyramid has a square base,
ABCD, and EF is the slant height.
computations.
E
Which statement is not true?
(1) EA::::: EC
(3) D.AEB ::::: D.BEC
(4) 6CED is isosceles
@EB:::::EF
¥J
16 What is the equation of a line passing through the point (6,1) and _
parallel to the line whose equation is 3x = 2y + 4?
~ X _
(1) y =
(2)
2
-3x + 5
2
y = -3x
-
3
~
@
(4)
3
2x -
8
3
y = 2x
-
5
y =
.
~
LJ
_.:.-
J_.-
-;::
y··-' - 3
~
; : :x
')
-~
Y' {t6rB
17 The volume of a sphere is approximately 44.6022 cubic centimeters.
What is the radius of the sphere, to the nearest tenth of a centimeter?
cw2.2
(3) 4.4
(2) 3.3
(4) 4.1
.
V~
u ~
-t I
I( 4. &o)-:J.. ,
/0, 6L!§' ~·
I v-
f"'r
r!>
J . J-2r
Geometry- June '13
[8]
3
3
.
0-fj /
y '; 1
J
f---t
18 Points A(5,3) and B(7,6) lie on AB . Points C(6,4) and D(9,0) lie on
CD. Which statement is true?
1'1 ~ ~· (; "]
(1)
AB) II CD
f---t
7- ) '/
'-flt'
f---t
1..
~
'f
lf-6
(2) AB l. CD
d f---t
1
fl1 CIS
~ ,: ;
(3)ABan CDarethesameine.
6-a
~~
f---t
)
1
~~B and CD intersect, but are not perpendicular.
f---t
Use this space for
computations.
c;
-
-1
>
19 Which set of equations represents two circles that have the same
center?
(1) x2
(2) (x
(3) (x
@ (x
+ (y + 4) 2 = 16 and (x + 4) 2 + y2 = 16
+ 3) 2 + (y - 3) 2 = 16 and (x - 3) 2 + (y + 3) 2 = 25
- 7) 2 + (y - 2) 2 = 16 and (x + 7) 2 + (y + 2) 2 = 25
- 2) 2 + (y - 5) 2 = 16 and (x - 2) 2 + (y - .5) 2 = 25
20 Transversal
below.
t;-:;::7
t./1
f---t
f---t
intersects AB and CD, as shown in the diagram
A
B
c
D
Which statement could always be used to prove AB
II CD?
(1) L2:::: L4
@ L3 and L6 are supplementary
(2) L1:::: L8
(4) L1 and L5 are supplementary
Geometry- June '13
[9]
[OVER]
AC>
21 In LiABC, mLA
inequality is true?
=
60, mLB
=
6C >AB
80, and mLC
~ A.B
(3) AC
(ij.}
(4)
> BC
AC > BC
=
Use this space for
computations.
40. \Vhich
< BA
BC < B1\
22 Circle 0 with LAOC and LABC is shown in the diagram below.
A
What is the ratio of mLAOC to mLABC?
).!( 1:1
\J.:V 2:1
(3) .3:1
(4) 1:2
23 A rectangular prism has a base \Vith 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 of the two prisms are equal, what is the height of the
second prism?
(1) 6
~ 12
(2) 8
(4) 15
lS" .
9.
)
>
~
24 In triangles ABC and DEF, AB = 4, AC = 5, DE = 8, DF = 10, and
LA :::::::
Which method could be used to prove LiABC ~ LiDEF?
fi·
(1) AN.
(3) SSS
@sAS
(4) ASA
Geometry -June '13
I) .: :.: 15.J~~ h
1), J-7o D
-~
[10]
,
-::
h
I
25 \Vhich graph represents a circle whose equation is x2
y
+ (y
Use this space for
computations.
- 1)2 = 9?
y
X
X
(3)
y
y
X
(2)
(4)
26 \Vhat is the perimeter of a rhombus whose diagonals are 16 and 30?
~ 92
(3) 60
~68
(4) 17
J]
17
Geometry- June '13
(11]
[OVER]
27 In right triangle ABC shown in the diagram below, altitude BD is
drawn to hypotenuse AC, CD = 12, and AD = 3.
Use this space for
computations.
A
}) · 3 ~X 'L
](;·~XJ
(;::X
8
!fi~J>~f{,'
c
~{f(S
What is the length of AB?
(1)
sJ3
@315
(2) 6
~
(4) 9
315
28 Secants JKL and JMN are drawn to circle 0 from an external point,].
IfJK = 8, LK = 4, and JM = 6, what is the length of JN?
((iY 16
(3) 10
(2) 12
(4) 8
V'}b ~ lAJb
f)-(B) >)·b
qG ~
G~
I&~ X
Geometry -June '13
[12]
Part II
Answer all 6 questions in this part. Each correct answer will receive 2 credits. Clearly
indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc. For all questions in this part, a correct numerical answer with no work shown will
receive only 1 credit. All answers should be written in pen, except for graphs and drawings,
which should be done in pencil. [12]
29 A right circular cylinder has a height of 7 inches and the base has a diameter of 6 inches.
Determine the lateral area, in square inches, of the cylinder in terms of '1T.
L~ :JJJrh
--J-1TOJL7)
:>
Geometry- June '13
LfJ-11
[13]
[OVER]
30 Determine, in degrees, the measure of each interior angle of a regular octagon.
~h
(?-~ /'10
---
~
I~ 5
Geometry -June •13
[14]
31 Triangle ABC has vertices at A(3,0), B(9, -5), and G(7, -8). Find the length of AC in simplest
radical form.
----;-:--:--
1{7-3))--1 C'i-v?
{/6+6y-
1f0
4:{5
Geometry- June '13
[15]
[OVER]
32 On the ray drawn below, using a compass and straightedge, construct an equilateral triangle with
a vertex at R. The length of a side of the triangle must be equal to a length of the diagonal of
rectangle ABCD.
D
c
A
B
R
Geometry- June '13
[16]
33 On the set of axes below, graph the locus of points 4 units from the x-axis and equidistant from
the points whose coordinates are ( -2,0) and (8,0).
Mark with an X all points that satisfy both conditions.
y
rI
I I
·+·············t··············+·············-+············1·········~··:-···············-~·············-t
!················f·············i··············+·············l··············+··········-·+················1··············4
.. ......
1
+-:.....:.
'\.~
~
.
i . ~- 1
i ·-+-f.-f--i
.........j-.
i- t=+· · · · · · ·t· · · · · · · +- · ·-·=~ == -~
Geometry- June '13
Rf·
····ll···········l··········-1 ·········!······+
[17]
I
-·-+.-~~+!~
r-·--+--1....--t----+'........-..J
· +·!-~· :~: ·: ·:.·~
i-······+·········+··········· ..•· .· · · · · · ·
.•·:·.··::.·:..
[OVER]
34 The coordinates of two vertices of square ABCD are A(2,1) and B(4,4).
Determine the slope of side BC.
'"-AS
(V}JO
Geometry -June '13
--
4 -1 ~J-
:L
~-;;..,
., ~).;
----3
[18]
Part III
Answer all 3 questions in this part. Each correct answer will receive 4 credits. Clearly
indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc. For all questions in this part, a correct numerical answer with no work shown will
receive only 1 credit. All answers should be written in pen, except for graphs and drawings,
which should be done in pencil. [12]
35 The coordinates of the vertices of parallelogram SWAN are S(2, -2), W( -2, -4), A( -4,6), and
N(0,8). State and label the coordinates of parallelogram S"W'A"N", the image of SWAN after the
transformation T4 _ 2 o D 1 . [The use of the set of axes below is optional.]
)' ( I , ~' ~t -I) ;)-) A'{-J
N >( 6/ ~)
<;''(_ s-,.,$)) ~/'{11-y)) ~}J{ ~!)) tJ"{41Jj
I
))
)
)
3)
I
y
X
Geometry -June '13
[19]
[OVER]
36 In circle 0 shown below, chords AB and CD and radius OA are drawn, such that AB:::: CD,
OE l. AB, OF l. CD, OF = 16, CF = y + 10, and CD = 4y - 20.
c
Determine the length of DF.
9y-Jo
J_{yrlo) ~
~
:J-y+)o
lfD "
J-0
Lfy .,)o
J-y
~y
Determine the length of OA.
6A
~DD ~
f;t]--+
~]
Geometry- June '13
lf
[20]
;t/
6fs
yf/()
~
d-0 1-) 0
~
30
37 If .6.RST- 6ABC, mLA = x2
-
8x, mLC
= 4x - 5, and mLR = 5x + 30, find mLC.
(Only an algebraic solution can receive full credit.]
X~--
?Y -~ >x-f') ()
X 7-. -.J 3~- ]()/ u
(x-!5) fyr)) ; o
:: r; {) ---s
s-
Geometry -June '13
s-s
[21]
[OVER]
Part IV
Answer the question in this part. A correct answer will receive 6 credits. Clearly
indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc. A correct numerical answer with no work shown will receive only 1 credit.
The answer should be written in pen, except for graphs and drawings, which should be done
in pencil. [6]
38 In the diagram of ./:').MAH below, MH ::::.:: AH and medians AB and MT are drawn.
Prove: LMBA ::::.:: LATM
1
Cl ve ">
:h
~-(!Qt.; vl- /Yof er ty
3 /}-efJ'II i-Ho h
t) (
;
soy_e)QS. fritJ)t<~
-lrlAhJJe J).wvU>~
OefJ~J,; }rtJn ()
Mol /tt~
/,cr;cJ-
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