Chapter 6.4, Problem 25PPS

To determine

The graph of each quadrilateral with given vertices.

Answer to Problem 25PPS

It is not a rectangle.

Explanation of Solution

Given:

The vertices are:

  $G\left(1,8\right),H\left(-7,7\right),J\left(-6,1\right),K\left(2,2\right)$

Calculation:

The graph will be:

The opposite sides of a parallelogram are congruent.

A rectangle has four right angles.

The slope of GH is:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{7-8}{-7-1}\\ m=\frac{-1}{-8}\\ m=\frac{1}{8}\end{array}$

The slope of HJ is:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{1-7}{-6-\left(-7\right)}\\ m=\frac{-6}{1}\\ m=-6\end{array}$

The slope of JK is:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{2-1}{2-\left(-6\right)}\\ m=\frac{1}{8}\end{array}$

The slope of GK is:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{2-8}{2-1}\\ m=\frac{-6}{1}\\ m=-6\end{array}$

The slope of GH is equal to slope of JK. The slope of HJ is equal to slope of GK. Hence, GH is parallel to JK and HJ is parallel to GK. Therefore, it is a parallelogram.

As, $-6\left(\frac{1}{8}\right)\ne -1,$ the adjacent sides of GHJK are not perpendicular.

Hence, it is not a rectangle.