Page 40 - Math Course 3 (Book 2)

Quadrilaterals: Coordinate Proofs

Coordinate Proof
By the Midpoint Formula, the coordinates
Proof:
of M, N, P, and Q are as follows.
Example
–2a + 0 0 –2b
M ( 2 , 2 = (–a, –b)
Write a coordinate proof to prove that the supports
2a + 0 0 + 2b of a platform lift are parallel.
P ( 2 , 2 = (a, b)
y
0 – 2a 2b + 0
N ( 2 , 2 = (–a, b) C(5, 10)

0 + 2a –2b + 0 D(0, 5) B(10, 5)
Q ( , = (a, –b)
2 2
Find the slopes of QP, MN, QM, and PN. O A(5, 0) X

slope of MQ = PN = 0 Given: A(5, 0), B(10, 5), C(5, 10), D(0, 5)

slope of QP = NM = undefined Prove: AB || CD

A segment with slope 0 is perpendicular to a seg- 5 – 0
ment with undefined slope. Therefore, consecutive Proof: slope of AB = or 1
10 – 5
sides of this quadrilateral are perpendicular. MNPQ 5 – 10
is, by definition, a rectangle. slope of CD = or 1
0 – 5
Since AB and CD have the same slope,
Your Turn! they are parallel.

Positioning a Square

Position and label a square with sides a units long on the coordinate plane. Which diagram would best
achieve this?

y y
A. B.
D(0, b) C(a, a) D(0, a) C(a, a)

O X O X
A(0, 0) B(b, 0) A(0, 0) B(a, 0)

C. y D. y
D(0, c) C(b, b) D(0, d) C(c, c)

O X O X
A(0, 0) B(a, 0) A(0, 0) B(b, 0)

Answer

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