Chapter 12.8, Problem 20E

To determine

To find:The correct assumption between the two friends.

Answer to Problem 20E

The friend with assumption that the quadrilateral is rectangle is correct.

Explanation of Solution

Given information:

  

A quadrilateral $PQRS$ with the vertices $P\left(0,2\right),Q\left(3,-4\right),R\left(1,5\right),S\left(-2,1\right)$ . One friend says the quadrilateral is parallelogram not a rectangle but the other friend says the other quadrilateral is rectangle.

If $PQRS$ is a rectangle then $PS\perp PQ$ . Therefore,

Evaluate the slope of $PS$ and $PQ$ . The slope is given by the formula,

  $m=\frac{y_2-y_1}{x_2-x_1}$

Here,

  $x_1=0,x_2=-2,y_1=2,y_2=1$

Now, the slope of $PS$ is given by substituting the value of $x_1,x_2,y_1,y_2$ in the given formula,

  $\begin{array}{c}{m}_1=\frac{1-2}{-2-0}\end{array}$

    $\begin{array}{c}=\frac{1}{2}\end{array}$

Now, the slope of $PQ$ is given by

  $\begin{array}{c}{m}_2=\frac{-4-2}{3-0}\end{array}$

    $\begin{array}{c}=-2\end{array}$

It can be seen that the slope of $PS$ is the reciprocal of negative of the slope of $PQ$ that is,

  $\begin{array}{c}{m}_1=\frac{-1}{-2}\end{array}$

    $\begin{array}{c}=\frac{1}{2}\end{array}$

So, the quadrilateral $PQRS$ is rectangle.

Therefore, the friend with assumption that the quadrilateral is rectangle is correct.