Question

the vertices of quadrilateral pqrs are listed.
p(3,7), q(6,-2), r(0,-4), s(-3,5)
which of the following is the strongest classification that identifies quadrilateral pqrs?
a. quadrilateral pqrs is a trapezoid.
b. quadrilateral pqrs is a rectangle.
c. quadrilateral pqrs is a square.
d. quadrilateral pqrs is a parallelogram.

Explanation:

Step1: Calculate slope of PQ

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{PQ}=\frac{-2-7}{6-3}=\frac{-9}{3}=-3$

Step2: Calculate slope of QR

$m_{QR}=\frac{-4-(-2)}{0-6}=\frac{-2}{-6}=\frac{1}{3}$

Step3: Calculate slope of RS

$m_{RS}=\frac{5-(-4)}{-3-0}=\frac{9}{-3}=-3$

Step4: Calculate slope of SP

$m_{SP}=\frac{7-5}{3-(-3)}=\frac{2}{6}=\frac{1}{3}$

Step5: Check parallel sides

$m_{PQ}=m_{RS}=-3$, so $PQ \parallel RS$; $m_{QR}=m_{SP}=\frac{1}{3}$, so $QR \parallel SP$. Both pairs of opposite sides are parallel, so it is a parallelogram.

Step6: Check perpendicularity (rectangle/square)

$m_{PQ} \times m_{QR} = -3 \times \frac{1}{3}=-1$, so $PQ \perp QR$. Now check side lengths:
Length of PQ: $\sqrt{(6-3)^2+(-2-7)^2}=\sqrt{9+81}=\sqrt{90}=3\sqrt{10}$
Length of QR: $\sqrt{(0-6)^2+(-4-(-2))^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10}$
Sides are not equal, so not a square.

Step7: Rule out trapezoid

Trapezoid has only one pair of parallel sides, but we have two pairs, so not a trapezoid.

Answer:

D. Quadrilateral PQRS is a parallelogram.