Question

quadrilateral $abcd$ has the following vertices:

• $a(-6, -2)$
• $b(-4,4)$
• $c(8,1)$
• $d(6,-6)$

also, $\angle a$ is a right angle.
is quadrilateral $abcd$ a rectangle, and why?
choose 1 answer:

Explanation:

Step1: Calculate slope of AB

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $A(-6,-2)$ and $B(-4,4)$, $m_{AB}=\frac{4 - (-2)}{-4-(-6)}=\frac{6}{2} = 3$.

Step2: Calculate slope of AD

For points $A(-6,-2)$ and $D(6,-6)$, $m_{AD}=\frac{-6 - (-2)}{6-(-6)}=\frac{-4}{12}=-\frac{1}{3}$. Since $\angle A$ is a right - angle, $m_{AB}\times m_{AD}=3\times(-\frac{1}{3})=- 1$.

Step3: Calculate slope of BC

For points $B(-4,4)$ and $C(8,1)$, $m_{BC}=\frac{1 - 4}{8-(-4)}=\frac{-3}{12}=-\frac{1}{4}$.

Step4: Calculate slope of CD

For points $C(8,1)$ and $D(6,-6)$, $m_{CD}=\frac{-6 - 1}{6 - 8}=\frac{-7}{-2}=\frac{7}{2}$.

Step5: Check parallel and perpendicular sides

In a rectangle, opposite sides are parallel (same slope) and adjacent sides are perpendicular (product of slopes is - 1). Since $m_{AB}
eq m_{CD}$ and $m_{BC}
eq m_{AD}$, the opposite sides are not parallel.

Answer:

No, because opposite sides are not parallel.