Chapter 6.6, Problem 13PPS

To determine

Whether the figure with given vertices is a trapezoid.

Answer to Problem 13PPS

JKLM is a trapezoid.

The lengths are not equal, hence JKLM is not an isosceles trapezoid.

Explanation of Solution

Given:

The given vertices are:

  $J\left(-4,-6\right),K\left(6,2\right),L(1,3),M\left(-4,-1\right).$

Calculation:

Draw the quadrilateral using given vertices:

  

A trapezoid has exactly one pair of opposite sides.

Use the slope formula to compare the opposite sides:

The slope of JK will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{2-\left(-6\right)}{6-\left(-4\right)}\\ m=\frac{8}{10}\\ m=\frac{4}{5}\end{array}$

The slope of KL will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{3-2}{1-6}\\ m=\frac{1}{-5}\\ m=-\frac{1}{5}\end{array}$

The slope of LM will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-1-3}{-4-1}\\ m=\frac{-4}{-5}\\ m=\frac{4}{5}\end{array}$

The slope of MJ will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-6-\left(-1\right)}{-4-\left(-4\right)}\\ m=\frac{-5}{0}\\ m=undefined\end{array}$

The slopes of exactly one pair of opposite sides are equal, so exactly one pair of opposite sides is parallel.

Hence, JKLM is a trapezoid.

Now, apply the Distance Formula to compare the lengths:

The length of KL will be:

  $\begin{array}{l}KL=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ KL=\sqrt{{\left(1-6\right)}^2+3-2^2}\\ KL=\sqrt{25+1}\\ KL=\sqrt{26}\end{array}$

The length of JM will be:

  $\begin{array}{l}JM=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ JM=\sqrt{{\left(-4-\left(-4\right)\right)}^2+{\left(-1-\left(-6\right)\right)}^2}\\ JM=\sqrt{0+25}\\ JM=\sqrt{25}\end{array}$

The lengths are not equal, hence JKLM is not an isosceles trapezoid.