Regents Exam Questions Name: ________________________
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
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1
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
1 Quadrilateral ABCD is graphed on the set of axes
below.
Which quadrilateral best classifies ABCD?
1) trapezoid
2) rectangle
3) rhombus
4) square
2 Square LMNO is shown in the diagram below.
What are the coordinates of the midpoint of
diagonal
LN
?
1)
4
1
2
,−2
1
2
2)
−3
1
2
,3
1
2
3)
−2
1
2
,3
1
2
4)
−2
1
2
,4
1
2
Regents Exam Questions Name: ________________________
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
www.jmap.org
2
3 In the diagram below, quadrilateral ABCD has
vertices
A(−5,1)
,
B(6, −1)
,
C(3, 5)
, and
D(−2,7)
.
What are the coordinates of the midpoint of
diagonal
AC
?
1)
(
−
1
,
3
)
2)
1
,
3
)
3)
1
,
4
)
4)
2
,
3
)
4 In the diagram below, parallelogram ABCD has
vertices
A(1,3)
,
B(5,7)
,
C(10, 7)
, and
D(6, 3)
.
Diagonals
AC
and
BD
intersect at E.
What are the coordinates of point E?
1)
(
0
.
5
,
2
)
2)
(
4
.
5
,
2
)
3)
(
5
.
5
,
5
)
4)
(
7
.
5
,
7
)
5 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
2)
AB
and
BC
3)
AD
and
BC
4)
AC
and
BD
6 Parallelogram ABCD has coordinates
A(1,5)
,
B(6,3)
,
C(3, −1)
, and
D(−2,1)
. What are the
coordinates of E, the intersection of diagonals
AC
and
BD
?
1)
2
,
2
)
2)
(
4
.
5
,
1
)
3)
(
3
.
5
,
2
)
4)
(
−
1
,
3
)
7 The coordinates of quadrilateral ABCD are
A(−1,−5)
,
B(8,2)
,
C(11,13)
, and
D(2, 6)
. Using
coordinate geometry, prove that quadrilateral
ABCD is a rhombus. [The use of the grid is
optional.]
Regents Exam Questions Name: ________________________
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
www.jmap.org
3
8 Given:
A(1,6), B(7, 9), C(13,6), and D(3, 1)
Prove: ABCD is a trapezoid. [The use of the
accompanying grid is optional.]
9 Ashanti is surveying for a new parking lot shaped
like a parallelogram. She knows that three of the
vertices of parallelogram ABCD are
A(0,0)
,
B(5,2)
,
and
C(6, 5)
. Find the coordinates of point D and
sketch parallelogram ABCD on the accompanying
set of axes. Justify mathematically that the figure
you have drawn is a parallelogram.
10 Quadrilateral ABCD has vertices
A(2,3)
,
B(7,10)
,
C(9, 4)
, and
D(4, −3)
. Prove that ABCD is a
parallelogram but not a rhombus. [The use of the
grid is optional.]
11 Given:
A(−2,2)
,
B(6,5)
,
C(4, 0)
,
D(−4,−3)
Prove: ABCD is a parallelogram but not a
rectangle. [The use of the grid is optional.]
Regents Exam Questions Name: ________________________
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
www.jmap.org
4
12 Quadrilateral MATH has coordinates
M(1, 1)
,
A(−2,5)
,
T(3, 5)
, and
H(6, 1)
. Prove that
quadrilateral MATH is a rhombus and prove that it
is not a square. [The use of the grid is optional.]
13 Jim is experimenting with a new drawing program
on his computer. He created quadrilateral TEAM
with coordinates
T(−2,3)
,
E(−5,−4)
,
A(2,−1)
, and
M(5, 6)
. Jim believes that he has created a rhombus
but not a square. Prove that Jim is correct. [The
use of the grid is optional.]
14 The vertices of quadrilateral JKLM have
coordinates
J(−3,1)
,
K(1, −5)
,
L(7, −2)
, and
M(3, 4)
.
Prove that JKLM is a parallelogram. Prove that
JKLM is not a rhombus. [The use of the set of axes
below is optional.]
Regents Exam Questions Name: ________________________
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
www.jmap.org
5
15 Quadrilateral KATE has vertices
K(1, 5),
A(4,7)
,
T(7, 3)
, and
E(1, −1)
.
a Prove that KATE is a trapezoid. [The use of the
grid is optional.]
b Prove that KATE is not an isosceles trapezoid.
16 The coordinates of quadrilateral JKLM are
J(1,−2)
,
K(13, 4)
,
L(6, 8)
, and
M(−2, 4)
. Prove that
quadrilateral JKLM is a trapezoid but not an
isosceles trapezoid. [The use of the grid is
optional.]
17 Given:
T(
−
1,1)
,
R(3,4)
,
A(7,2)
, and
P(
−
1,
−
4)
Prove: TRAP is a trapezoid.
TRAP is not an isosceles trapezoid.
[The use of the grid is optional.]
18 Given: Quadrilateral ABCD has vertices
A(−5,6)
,
B(6,6)
,
C(8, −3)
, and
D(−3,−3)
.
Prove: Quadrilateral ABCD is a parallelogram but
is neither a rhombus nor a rectangle. [The use of
the grid below is optional.]
Regents Exam Questions Name: ________________________
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
www.jmap.org
6
19 Quadrilateral ABCD with vertices
A(
−
7,4)
,
B(−3,6)
,
C(3, 0)
, and
D(1, −8)
is graphed on the set
of axes below. Quadrilateral MNPQ is formed by
joining M, N, P, and Q, the midpoints of
AB
,
BC
,
CD
, and
AD
, respectively. Prove that quadrilateral
MNPQ is a parallelogram. Prove that quadrilateral
MNPQ is not a rhombus.
20 Given:
ABC
with vertices
A(−6,−2)
,
B(2,8)
, and
C(6, −2)
.
AB
has midpoint D,
BC
has midpoint E,
and
AC
has midpoint F.
Prove: ADEF is a parallelogram
ADEF is not a rhombus
[The use of the grid is optional.]
21 In the accompanying diagram of ABCD, where
a ≠ b
, prove ABCD is an isosceles trapezoid.
22 The coordinates of quadrilateral PRAT are
P(a,b)
,
R(a,b + 3),
A(a + 3, b + 4)
, and
T(a + 6, b + 2)
.
Prove that
RA
is parallel to
PT
.
23 The coordinates of two vertices of square ABCD
are
A(2,1)
and
B(4,4)
. Determine the slope of side
BC
.
24 Rectangle KLMN has vertices
K(0, 4)
,
L(4, 2)
,
M(1, −4)
, and
N(−3,−2)
. Determine and state the
coordinates of the point of intersection of the
diagonals.
ID: A
1
G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2
Answer Section
1 ANS: 3
Both pairs of opposite sides are parallel, so not a trapezoid. None of the angles are right angles, so not a rectangle
or square. All sides are congruent, so a rhombus.
REF: 081411ge
2 ANS: 4
M
x
=
−6 + 1
2
= −
5
2
.
M
y
=
1 + 8
2
=
9
2
.
REF: 060919ge
3 ANS: 1
M
x
=
−5 + 3
2
=
−2
2
= −1
.
M
y
=
1 + 5
2
=
6
2
= 3
.
REF: 061402ge
4 ANS: 3
M
x
=
1 + 10
2
=
11
2
= 5.5
M
y
=
3 + 7
2
=
10
2
= 5
.
REF: 081407ge
5 ANS: 2
Adjacent sides of a rectangle are perpendicular and have opposite and reciprocal slopes.
REF: 061028ge
6 ANS: 1
The diagonals of a parallelogram intersect at their midpoints.
M
AC
1 + 3
2
,
5 + (−1)
2
= (2,2)
REF: 061209ge
ID: A
2
7 ANS:
. To prove that ABCD is a rhombus, show that all sides are congruent using the distance formula:
d
AB
= (8 − (−1))
2
+ (2 − (−5))
2
= 130
d
BC
= (11 − 8)
2
+ (13 − 2)
2
= 130
d
CD
= (11 − 2)
2
+ (13 − 6)
2
= 130
d
AD
= (2 − (−1))
2
+ (6 − (−5))
2
= 130
.
REF: 060327b
8 ANS:
. To prove that ABCD is a trapezoid, show that one pair of opposite sides of the figure is parallel
by showing they have the same slope and that the other pair of opposite sides is not parallel by showing they do
not have the same slope:
m
AB
=
9 − 6
7 − 1
=
3
6
=
1
2
m
CD
=
6 − 1
13 − 3
=
5
10
=
1
2
m
AD
=
6 − 1
1 − 3
= −
5
2
m
BC
=
9 − 6
7 − 13
= −
3
6
= −
1
2
REF: 080134b
ID: A
3
9 ANS:
Both pairs of opposite sides of a parallelogram are parallel. Parallel
lines have the same slope. The slope of side
BC
is 3. For side
AD
to have a slope of 3, the coordinates of point D
must be
(1,3)
.
m
AB
=
2 − 0
5 − 0
=
2
5
m
CD
=
5 − 3
6 − 1
=
2
5
m
AD
=
3 − 0
1 − 0
= 3
m
BC
=
5 − 2
6 − 5
= 3
REF: 080032a
10 ANS:
m
AB
=
10 − 3
7 − 2
=
7
5
,
m
CD
=
4 − (−3)
9 − 4
=
7
5
,
m
AD
=
3 − (−3)
2 − 4
=
6
−2
= −3
,
m
BC
=
10 − 4
7 − 9
=
6
−2
= −3
(Definition of
slope).
AB CD
,
AD BC
(Parallel lines have equal slope). Quadrilateral ABCD is a parallelogram (Definition of
parallelogram).
d
AD
= (2 − 4)
2
+ (3 − (−3))
2
= 40
,
d
AB
= (7 − 2)
2
+ (10 − 3)
2
= 74
(Definition of
distance).
AD
is not congruent to
AB
(Congruent lines have equal distance). ABCD is not a rhombus (A rhombus
has four equal sides).
REF: 061031b
ID: A
4
11 ANS:
To prove that ABCD is a parallelogram, show that both pairs of opposite sides of the parallelogram are parallel by
showing the opposite sides have the same slope:
m
AB
=
5 − 2
6 − (−2)
=
3
8
m
CD
=
−3 − 0
−4 − 4
=
3
8
m
AD
=
−3 − 2
−4 − (−2)
=
5
2
m
BC
=
5 − 0
6 − 4
=
5
2
A rectangle has four right angles. If ABCD is a rectangle, then
AB ⊥ BC
,
BC ⊥ CD
,
CD ⊥ AD
, and
AD ⊥ AB
.
Lines that are perpendicular have slopes that are the opposite and reciprocal of each other. Because
3
8
and
5
2
are
not opposite reciprocals, the consecutive sides of ABCD are not perpendicular, and ABCD is not a rectangle.
REF: 060633b
12 ANS:
The length of each side of quadrilateral is 5. Since each side is congruent, quadrilateral
MATH is a rhombus. The slope of
MH
is 0 and the slope of
HT
is
−
4
3
. Since the slopes are not negative
reciprocals, the sides are not perpendicular and do not form rights angles. Since adjacent sides are not
perpendicular, quadrilateral MATH is not a square.
REF: 011138ge
ID: A
5
13 ANS:
. To prove that TEAM is a rhombus, show that all sides are congruent using the distance formula:
d
ET
=(−2 − (−5))
2
+ (3 − (−4))
2
= 58
d
AM
= (2 − 5)
2
+ ((−1) − 6)
2
= 58
d
AE
=(−5 − 2)
2
+ (−4 − (−1))
2
= 58
d
MT
=(−2 − 5)
2
+ (3 − 6)
2
= 58
. A square has four right angles. If TEAM is a square, then
ET ⊥ AE
,
AE ⊥ AM
,
AM ⊥ AT
and
MT ⊥ ET
. Lines that are perpendicular have slopes that are opposite reciprocals of each
other. The slopes of sides of TEAM are:
m
ET
=
−4 − 3
−5 − (−2)
=
7
3
m
AM
=
6 − (−1)
5 − 2
=
7
3
m
AE
=
−4 − (−1)
−5 − 2
=
3
7
m
MT
=
3 − 6
−2 − 5
=
3
7
Because
7
3
and
3
7
are not
opposite reciprocals, consecutive sides of TEAM are not perpendicular, and TEAM is not a square.
REF: 010533b
14 ANS:
m
JM
=
1 − 4
−3 − 3
=
−3
−6
=
1
2
m
= ML
=
4 −−2
3 − 7
=
6
−4
= −
3
2
m
LK
=
−2 −−5
7 − 1
=
3
6
=
1
2
m
KJ
=
−5 − 1
1 −−3
=
−6
4
= −
3
2
Since both opposite sides have equal slopes and are parallel, JKLM is a parallelogram.
JM = (−3 − 3)
2
+ (1 − 4)
2
= 45
ML = (7 − 3)
2
+ (−2 − 4)
2
= 52
.
JM
is not congruent to
ML
, so JKLM is not a rhombus since not all sides
are congruent.
REF: 061438ge
ID: A
6
15 ANS:
. To prove that KATE is a trapezoid, show that one pair of opposite sides of the figure is parallel by
showing they have the same slope and that the other pair of opposite sides is not parallel by showing they do not
have the same slope:
m
AK
=
7 − 5
4 − 1
=
2
3
m
ET
=
3 − (−1)
7 − 1
=
4
6
=
2
3
m
EK
=
−1 − 5
1 − 1
= undefined
m
AT
=
7 − 3
4 − 7
= −
4
3
To prove that a trapezoid is not an isosceles trapezoid, show that the opposite sides that are not parallel are also
not congruent using the distance formula:
d
EK
= (1 − 1)
2
+ (5 − (−1))
2
= 6
d
AT
= (4 − 7)
2
+ (7 − 3)
2
= 5
REF: 010333b
16 ANS:
. To prove that JKLM is a trapezoid, show that one pair of opposite sides of the figure is parallel
by showing they have the same slope and that the other pair of opposite sides is not parallel by showing they do
not have the same slope:
m
JK
=
4 − (−2)
13 − 1
=
1
2
m
LM
=
8 − 4
6 − (−2)
=
1
2
m
JM
=
−2 − 4
1 − (−2)
= −2
m
KL
=
4 − 8
13 − 6
= −
4
7
To prove that a trapezoid is not an isosceles trapezoid, show that the opposite sides that are not parallel are also
not congruent using the distance formula:
d
JM
= (1 − (−2))
2
+ (−2 − 4)
2
= 45
d
KL
= (13 − 6)
2
+ (4 − 8)
2
= 65
REF: 080434b
ID: A
7
17 ANS:
. To prove that TRAP is a trapezoid, show that one pair of opposite sides of the figure is
parallel by showing they have the same slope and that the other pair of opposite sides is not parallel by showing
they do not have the same slope:
m
TR
=
1 − 4
−1 − 3
=
3
4
m
PA
=
−4 − 2
−1 − 7
=
3
4
m
TP
=
1 − (−4)
−1 − (−1)
= undefined
m
RA
=
4 − 2
3 − 7
= −
1
2
To prove that a trapezoid is not an isosceles trapezoid, show that the opposite sides that are not parallel are also
not congruent using the distance formula:
d
TP
= (−1 − (−1))
2
+ (1 − (−4))
2
= 5
d
RA
= (3 − 7)
2
+ (4 − 2)
2
= 20 = 25
REF: 080933b
18 ANS:
AB CD
and
AD CB
because their slopes are equal. ABCD is a parallelogram
because opposite side are parallel.
AB ≠ BC .
ABCD is not a rhombus because all sides are not equal.
AB ∼⊥BC
because their slopes are not opposite reciprocals. ABCD is not a rectangle because
∠ABC
is not a
right angle.
REF: 081038ge
ID: A
8
19 ANS:
M
−7 +−3
2
,
4 + 6
2
= M(−5,5)
N
−3 + 3
2
,
6 + 0
2
= N(0, 3)
P
3 + 1
2
,
0 +−8
2
= P(2,−4)
Q
−7 + 1
2
,
4 +−8
2
= Q(−3,−2)
.
m
MN
=
5 − 3
−5 − 0
=
2
−5
m
PQ
=
−4 −−2
2 −−3
=
−2
5
m
NA
=
3 −−4
0 − 2
=
7
−2
m
QM
=
−2 − 5
−3 −−5
=
−7
2
. Since both opposite sides have equal slopes and are
parallel, MNPQ is a parallelogram.
MN = (−5 − 0)
2
+ (5 − 3)
2
= 29
NA = (0 − 2)
2
+ (3 −−4)
2
= 53
.
MN
is not congruent to
NP
, so MNPQ
is not a rhombus since not all sides are congruent.
REF: 081338ge
20 ANS:
m
AB
=
−6 + 2
2
,
−2 + 8
2
= D(2, 3)
m
BC
=
2 + 6
2
,
8 +−2
2
= E(4,3)
F(0, −2)
. To prove that ADEF is a
parallelogram, show that both pairs of opposite sides of the parallelogram are parallel by showing the opposite
sides have the same slope:
m
AD
=
3 −−2
−2 −−6
=
5
4
m
FE
=
3 −−2
4 − 0
=
5
4
AF DE
because all horizontal lines have the same slope. ADEF
is not a rhombus because not all sides are congruent.
AD = 5
2
+ 4
2
= 41
AF = 6
REF: 081138ge
ID: A
9
21 ANS:
To prove that ABCD is a trapezoid, show that one pair of opposite sides of the figure is parallel by showing they
have the same slope and that the other pair of opposite sides is not parallel by showing they do not have the same
slope:
m
AB
=
0 − 0
−a − a
=
0
−2a
= 0
m
CD
=
c − c
−b − b
=
0
−2b
= 0
m
AD
=
c − 0
−b − (−a)
=
c
−b + a
m
BC
=
c − 0
b − a
=
c
b − a
If
AD
and
BC
are parallel, then:
c
−b + a
=
c
b − a
c(b − a) = c(−b + a)
b − a = −b + a
2a = 2b
a = b
But the facts of the problem indicate
a ≠ b
, so
AD
and
BC
are not parallel.
To prove that a trapezoid is an isosceles trapezoid, show that the opposite sides that are not parallel are congruent
using the distance formula:
d
BC
= (b − a)
2
+ (c − 0)
2
= b
2
− 2ab + a
2
+ c
2
= a
2
+ b
2
− 2ab + c
2
d
AD
= (−b − (−a))
2
+ (c − 0)
2
= (a − b)
2
+ c
2
= a
2
− 2ab + b
2
+ c
2
= a
2
+ b
2
− 2ab + c
2
REF: 080534b
22 ANS:
m
RA
=
(b + 3) − (b + 4)
a − (a + 3)
=
−1
−3
=
1
3
m
PT
=
b − (b + 2)
a − (a + 6)
=
−2
−6
=
1
3
. Because
RA
and
PT
have equal slopes, they are parallel.
REF: 060824b
23 ANS:
m
AB
=
4 − 1
4 − 2
=
3
2
.
m
BC
= −
2
3
REF: 061334ge
24 ANS:
0 + 1
2
,
4 +−4
2
1
2
,0
REF: 081534ge