Preparing for the Regents Examination Geometry, AK

5 BIG is isosceles because it has two congruent
sides, BI IG √ 50 ; and BIG is
a right triangle because −−
BI is perpendicular
to −−
IG .
6 Two sides ( −−−
QR and −−
PS ) have the same slope
of 2 _ and the other two sides (
3 −−
OP and −−
RS )
have the same slope of 3. PQRS is a
parallelogram.
7 SAND is a parallelogram because the diagonals
bisect each other at the point (1.5, 0.5).
8 LEAP is a parallelogram because all of the
sides measure √ 50 and are congruent.
9 ABCD is a parallelogram because the diagonals
bisect each other at the point (2.5, 0.5).
10 a 6 √ 5 and 2 √ 13
b The diagonals meet at their midpoint
(1, 1).
c BETH is a parallelogram because the
diagonals bisect each other.
d BETH is not a rectangle because the
diagonals are not congruent.
11 a NICK is a parallelogram; both pairs of
opposite sides are parallel.
b NICK is not a rhombus; the diagonals are
not perpendicular.
12 The figure KATE is a square. KATE is a rectangle
because the diagonals bisect each other
and are the same length. KATE is a square
because two adjacent sides are congruent.
13 ABCD is a rhombus. ABCD is a parallelogram
because the diagonals bisect each other.
ABCD is a rhombus because two adjacent
sides are congruent.
14 a Slopes of −−
SU and −−
UE are negative reciprocals
proving these two lines form a right
angle by being perpendicular; therefore,
SUE is a right triangle.
b SU 5 and UE 10
15 a NORA is a parallelogram because the
diagonals bisect each other. NORA is a
rhombus because two adjacent sides are
congruent.
b NORA is not a square because it does not
contain a right angle.
42 Chapter 8: Slopes and Equations of Lines
16 a JACK is a trapezoid because it has only
one pair of opposite sides parallel;
slope of −−
JA slope of −−
CR 1.
b JACK is not an isosceles trapezoid because
the legs are not congruent.
17 MARY is a trapezoid because it has only
one pair of opposite sides parallel; slope of
−−−
MA slope of −−
RY 0. MARY is an isosceles
trapezoid because the legs are congruent;
AR YM 5.
18 C(a, 0)
19 P(b a, c)
20 E(a, 0), F(a, 2a), G(a, 2a)
21 C(a, b r)
22 M(a c, b)
23 T(a, 2a)
24 R(a b, c)
25 Yes
26 Slope of −−
AC is 1 and the slope of −−
BD is 1.
Since the slopes are negative reciprocals,
−−
AC −−
BD .
27 a Midpoint of −−
BC is (a, b).
28 a
b MA MB MC √ a 2 b 2
T(0, 0)
y
E(2x, 2y)
Q(4x, 0)
x
b A(x, y), B(3x, y), C(2x, 0)
c The length of median −−
TB is √ 9 x 2 y 2 .
The length of median −−
EC is 2y. The length
of median −−−
QA is √ 9 x 2 y 2 .
d TEQ is an isosceles triangle.
29 TA TR √
( c _
2 ) 2
d 2
30 −−
JA and −−
NE are parallel to the x-axis and
each slope is 0; thus they are parallel to each
other. Slope of −−−
AN slope of −−
EJ c _ ; thus
b
they are parallel. Therefore, JANE is a
parallelogram.
31 Midpoint of −−
AC midpoint of
−−
BD
a r ( _ ,
s
2 _
2 )
32 TA EM √
(a b) 2 c 2