Question

parallelogram $abcd$ has the following vertices:
$a(0,8)$
$b(8,4)$
$c(2, - 8)$
$d(-6,-4)$
is parallelogram $abcd$ a rectangle, and why?
choose 1 answer:

Explanation:

Step1: Find the slopes of adjacent sides

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Slope of $AB$: $m_{AB}=\frac{4 - 8}{8 - 0}=\frac{-4}{8}=-\frac{1}{2}$
Slope of $BC$: $m_{BC}=\frac{-8 - 4}{2 - 8}=\frac{-12}{-6}=2$
Slope of $CD$: $m_{CD}=\frac{-4+8}{-6 - 2}=\frac{4}{-8}=-\frac{1}{2}$
Slope of $DA$: $m_{DA}=\frac{8 + 4}{0+6}=\frac{12}{6}=2$

Step2: Check if adjacent - sides are perpendicular

Two lines are perpendicular if the product of their slopes is $- 1$.
$m_{AB}\times m_{BC}=-\frac{1}{2}\times2=-1$
$m_{BC}\times m_{CD}=2\times(-\frac{1}{2})=-1$
$m_{CD}\times m_{DA}=-\frac{1}{2}\times2=-1$
$m_{DA}\times m_{AB}=2\times(-\frac{1}{2})=-1$

Answer:

Yes, because the product of the slopes of adjacent sides is - 1, which means adjacent sides are perpendicular.