Regents Exam Questions Name: ________________________
G.GPE.B.4: Triangles in the Coordinate Plane
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1
G.GPE.B.4: Triangles in the Coordinate Plane
1 On the set of axes below,
ABC
, altitude
CG
, and
median
CM
are drawn.
Which expression represents the area of
ABC
?
1)
(BC)(AC)
2
2)
(GC)(BC)
2
3)
(CM)(AB)
2
4)
(GC)(AB)
2
2 In the diagram below,
ABC
has vertices
A(4,5)
,
B(2,1)
, and
C(7, 3)
.
What is the slope of the altitude drawn from A to
BC
?
1)
2
5
2)
3
2
3)
−
1
2
4)
−
5
2
3 The coordinates of the vertices of
RST
are
R(−2,−3)
,
S(8, 2)
, and
T(4, 5)
. Which type of
triangle is
RST
?
1) right 2) acute 3) obtuse 4) equiangular
4 Triangle ABC has vertices
A(0,0)
,
B(3,2)
, and
C(0, 4)
. The triangle may be classified as
1) equilateral 2) isosceles 3) right
4) scalene
5 Which type of triangle can be drawn using the
points
(−2,3)
,
(−2,−7)
, and
(4,−5)
?
1) scalene 2) isosceles 3) equilateral 4) no
triangle can be drawn
6 If the vertices of
ABC
are
A(−2,4)
,
B(−2,8)
, and
C(−5,6)
, then
ABC
is classified as
1) right 2) scalene 3) isosceles
4) equilateral
7 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
Regents Exam Questions Name: ________________________
G.GPE.B.4: Triangles in the Coordinate Plane
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2
8 On the set of axes below,
DEF
has vertices at the
coordinates
D(1, −1)
,
E(3,4)
, and
F(4,2)
, and point
G has coordinates
(3,1)
. Owen claims the median
from point E must pass through point G. Is Owen
correct? Explain why.
9 Triangle RST has vertices with coordinates
R(−3,−2)
,
S(3, 2)
and
T(4, −4)
. Determine and state
an equation of the line parallel to
RT
that passes
through point S. [The use of the set of axes below
is optional.]
10 Given: Triangle RST has coordinates
R(
−
1,7)
,
S(3, −1)
, and
T(9, 2)
Prove:
RST
is a right triangle
[The use of the set of axes below is optional.]
Regents Exam Questions Name: ________________________
G.GPE.B.4: Triangles in the Coordinate Plane
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3
11 Triangle ABC has vertices with
A(x, 3)
,
B(
−
3,
−
1)
,
and
C(−1,−4)
. Determine and state a value of x
that would make triangle
ABC
a right triangle.
Justify why
ABC
is a right triangle. [The use of
the set of axes below is optional.]
12 Given:
J(−4,1)
,
E(−2,−3)
,
N(2, −1)
Prove:
JEN
is an isosceles right triangle.
[The use of the grid is optional.]
13 A triangle has vertices
A(
−
2,4)
,
B(6,2)
, and
C(1, −1)
. Prove that
ABC
is an isosceles right
triangle. [The use of the set of axes below is
optional.]
14 Triangle JOE has vertices whose coordinates are
J(4,6)
,
O(−2,4)
, and
E(6,0)
. Prove that
JOE
is
isosceles. Point
Y(2, 2)
is on
OE
. Prove that
JY
is
the perpendicular bisector of
OE
. [The use of the
set of axes below is optional.]
Regents Exam Questions Name: ________________________
G.GPE.B.4: Triangles in the Coordinate Plane
www.jmap.org
4
15 Triangle ABC has vertices with coordinates
A(−1,−1)
,
B(4,0)
, and
C(0, 4)
. Prove that
ABC
is
an isosceles triangle but not an equilateral triangle.
[The use of the set of axes below is optional.]
16 Triangle PQR has vertices
P(
−
3,
−
1)
,
Q(
−
1,7)
, and
R(3,3)
, and points A and B are midpoints of
PQ
and
RQ
, respectively. Use coordinate geometry to
prove that
AB
is parallel to
PR
and is half the
length of
PR
. [The use of the set of axes below is
optional.]
ID: A
1
G.GPE.B.4: Triangles in the Coordinate Plane
Answer Section
1 ANS: 4 REF: 011921geo
2 ANS: 4
The slope of
BC
is
2
5
. Altitude is perpendicular, so its slope is
−
5
2
.
REF: 061614geo
3 ANS: 1
m
RT
=
5 −−3
4 −−2
=
8
6
=
4
3
m
ST
=
5 − 2
4 − 8
=
3
−4
= −
3
4
Slopes are opposite reciprocals, so lines form a right angle.
REF: 011618geo
4 ANS: 2 REF: 061115ge
5 ANS: 2 REF: 081226ge
6 ANS: 3
AB = 8 − 4 = 4
.
BC = (−2 − (−5))
2
+ (8 − 6)
2
= 13
.
AC = (−2 − (−5))
2
+ (4 − 6)
2
= 13
REF: 011328ge
7 ANS: 1
Since
AC ≅ BC
,
m∠A = m∠B
under the Isosceles Triangle Theorem.
REF: fall0809ge
8 ANS:
No. The midpoint of
DF
is
1 + 4
2
,
−1 + 2
2
= (2.5, 0.5)
. A median from point E must pass through the midpoint.
REF: 011930geo
9 ANS:
−2 −−4
−3 − 4
=
2
−7
;
y − 2 = −
2
7
(x − 3)
REF: 062331geo
ID: A
2
10 ANS:
REF: 011637ge
11 ANS:
The slopes of perpendicular line are opposite reciprocals. Since the lines are perpendicular, they form right angles
and a right triangle.
m
BC
= −
3
2
m
⊥
=
2
3
−1 =
2
3
(−3) + b
−1 = −2 + b
1 = b
3 =
2
3
x + 1
2 =
2
3
x
3 = x
or
−4 =
2
3
(−1) + b
−12
3
=
−2
3
+ b
−
10
3
= b
3 =
2
3
x −
10
3
9 = 2x − 10
19 = 2x
9.5 = x
REF: 081533geo
ID: A
3
12 ANS:
To prove that
JEN
is a right triangle, prove that its legs are perpendicular by showing their
slopes are opposite reciprocals:
m
JE
=
1 −−3
−4 −−2
=
4
−2
= −2
m
EN
=
−3 −−1
−2 − 2
=
−2
−4
=
1
2
To prove that
JEN
is an isosceles triangle, prove that it legs are congruent by using the distance formula:
d
JE
= (−4 − (−2))
2
+ (1 − (−3))
2
= 20
d
EN
= (−2 − 2)
2
+ (−3 −−1)
2
= 20
REF: 011029b
13 ANS:
Triangle with vertices
A(−2,4)
,
B(6,2)
, and
C(1, −1)
(given);
m
AC
= −
5
3
,
m
BC
=
3
5
,
definition of slope; Because the slopes of the legs of the triangle are opposite reciprocals, the legs are
perpendicular (definition of perpendicular);
∠C
is a right angle (definition of right angle);
ABC
is a right
triangle (if a triangle has a right angle, it is a right triangle);
AC ≅ BC = 34
(distance formula);
ABC
is an
isosceles triangle (an isosceles triangle has two congruent sides).
REF: 011932geo
ID: A
4
14 ANS:
JE = JO = 6
2
+ 2
2
= 40
Since
JOE
has two congruent sides, it is isosceles.
OY = YE = 4
2
+ 2
2
= 20
Since
OY ≅ YE
,
JY
is a bisector of
OE
.
m
OE
=
4
−8
= −
1
2
m
JY
=
4
2
= 2
Since the
slopes are opposite reciprocals,
OE⊥JY
.
REF: 062435geo
15 ANS:
Because
AB ≅ AC
,
ABC
has two congruent sides and is isosceles. Because
AB ≅ BC
is not true,
ABC
has sides that are not congruent and
ABC
is not equilateral.
REF: 061832geo
16 ANS:
REF: 081732geo
