Name: ____________________
Date: ___________
CC Geometry R
HW #10
1) Complete:
equal
a) Parallel line have ________________________
slopes.
right
b) Perpendicular lines, which form _____________
angles have
opposite reciprocal
_______________ __________________ slopes.
equal
lengths
c) Congruent segments have ______________
__________________.
midpoint
d) Segments that bisect each other share the same _________________.
2) ΔAFN: A(-7,6), F(-1,6), N(-4,2). Prove the triangle is isosceles, but not
equilateral. Use of graph is optional. Label your work and write a concluding
statement.
Prove ΔAFN is isosceles:
AN = √(­7 + 4)2 + (6 ­ 2)2 = √ 9 + 16 = √25 = 5
FN = √(­1 + 4)2 + (6 ­ 2)2 = √9 + 16 = √25 = 5
AN = FN
Prove ΔAFN is not equilateral:
AF = 6
AF ≠ AN or FN
ΔAFN is isosceles because it has two equal sides, but it is not
equilateral because it does not have 3 equal sides.
F
A
N
2) Quadrilateral ABCD has vertices A(-1,5), B(5,1), C(6,-2) and D(0,2).
Prove that ABCD is a parallelogram.
Method I: (Show diagonals bisect each other.)
midpoint (AC) = (2.5, 1.5)
midpoint (BD) = (2.5, 1.5)
ABCD is a parallelogram because its diagonals bisect each other.
Method 2: (Show both pairs of opposite sides are parallel.)
slope (AD) = ­3, slope (BC) = ­3 AD ll BC
slope (AB) = ­ 2/3, slope (DC) = ­2/3 AB ll DC ABCD is a parallelogram because both pairs of
opposite sides are parallel.
Method 3: (Show both pairs of opposite sides are equal.)
AD = √10 , BC = √10 AD = BC
AB = √52, DC = √52 AB = DC ABCD is a parallelogram because both pairs of
opposite sides are equal in length.
Method 4: (Show one pair of sides is both = and ll.)
AD = √10 , BC = √10 AD = BC
slope of AD = ­3, slope of BC = ­3 AD ll BC ABCD is a parallelogram because one pair of
sides are both parallel and equal.
A
D
B
C
4) Prove that quadrilateral ABCD with: A(-1,4), B(2,6), C
(5,4) and D(2,2) is a rhombus.
Method 1: (Prove ABCD is equilateral.)
AD = √13, BC = √13, CD = √13, DA = √13
An equilateral quadrilateral is a rhombus.
B
A
C
Methods 2 and 3: First, prove ABCD is a parallelogram by one of the four methods : See Ex. (3).
D
Then prove one of the following:
(2) diagonals are perpendicular
slope of AC = 0 (horizontal line), slope of BD = undefined (vertical line)
AC BD
A parallelogram with perpendicular diagonasl is a rhombus. OR
(3) two adjacent sides are =
AB = √13, BC = √13 AB = BC
A parallelogram with two equal adjacent sides is a rhombus. 5) 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.
a) Prove that quadrilateral MNPQ is a parallelogram.
b) Prove that quadrilateral MNPQ is not a rhombus.
a)
(­3,6)
M
N
(­7,4)
(3,0)
Q
P
(1,­8)