Preparing for the Regents Examination Geometry, AK

3 4x 2 3x 3
x 5
RS 18
4 Perimeter of MABC 8
5 mADC 110
6 2x 2 x 8
x 10
CD 18
7 mADC 94
8 a Midpoint of −−
AC midpoint of −−
BD (6, 5)
b Slope of −−
AC 1, slope of −−
BD 1.
Slopes are negative reciprocals, diagonals
are perpendicular.
9 a (9, 5)
b PQ PS 5 √ 2
c Slope of −−
PR 1 _ , slope of
3 −−
QS 3.
Slopes are negative reciprocals, diagonals
are perpendicular.
10 a Slope of −−−
AD slope of −−
BC k _
x
.
AD BC √ x 2 k 2 . ABCD is a parallelogram.
(One pair of opposite sides have
the same length and are parallel.)
b Slope of −−
CA k _
−−
, slope of BD
x k k _
x k ,
not negative reciprocals. The diagonals are
not perpendicular.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
11 −−−
QR −−
AC , −−
PS −−
AC , −−−
QR −−
PS . Likewise,
−−
QP −−
BD , −−
RS −−
BD , and −−
QP −−
RS . Thus, PQRS
is a parallelogram. Since −−
AC −−
BD , −−−
QR
and −−
PS are perpendicular to −−
QP and −−
RS .
Thus, PQRS has at least one right angle, and
PQRS is a rectangle.
12 1. Rhombus ABCD
2. −−
ED
3. −−
BF −−
AC
4. −−
EF −−
AC
5. F bisects −−
AC .
6. EA EC
62 Chapter 10: Quadrilaterals
7. −−
EA −−
EC
8. ACE is isosceles. (Definition of an
isosceles triangle)
13 Prove that −−
DF −−
AB and −−
DE −−
BF by using
the midpoints and alternate interior angles.
Thus, EBFD is a parallelogram. By the division
postulate, −−
EB −−
FB . EBFD is a rhombus
because a rhombus is a parallelogram with
two congruent consecutive sides.
14 1. Parallelogram ABCD
2. −−
AB −−−
CD
3. BAD 2
4. mBAD m2
(Corresponding angles
are congruent.)
5. BAD 1 CAD
6. mBAD m1 (Whole is greater
than a part.)
7. m2 m1 (Substitution
postulate)
15 1. Rhombus PQRS
2. −−
QS −−
PR
3. −−−
QC −−
CS
4. B is the midpoint of −−−
QC .
5. D is the midpoint of −−
CS .
6. −−
BC −−−
CD
7. ___
AC ___
AC
8. ABC ADC
9.
(SAS SAS)
−−
AB −−−
AD
(CPCTC)
10. BAD is isosceles. (Definition of an
isosceles triangle)
16 If a quadrilateral is equilateral, it is a parallelogram
with a pair of consecutive congruent
sides.
17 The diagonal creates two congruent triangles
(ASA ASA) and corresponding congruent
sides are consecutive.
18 1. −−−
CD BE
2. −−−
CD −−
BA
3. −−−
AD BF
4. −−−
AD −−
BC
5. ABCD is a ( Definition of a
parallelogram. parallelogram)
6. BAD BCD (Opposite angles
7. BG bisects FBE.
are congruent.)
8. CBD ABD
9. ABCD is a rhombus. (Diagonal −−
BD
bisects opposite
angles.)