Chapter 6.6, Problem 3CYU

To determine

Whether the figure with given vertices is a trapezoid.

Answer to Problem 3CYU

ABCD is a trapezoid.

Explanation of Solution

Given:

The given vertices are:

  $A\left(-4,-1\right),B\left(-2,3\right),C(3,3),D\left(5,-1\right).$

Calculation:

Draw the quadrilateral using given vertices:

  

Use the slope formula to compare the opposite sides:

The slope of AB will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{3-\left(-1\right)}{-2-\left(-4\right)}\\ m=\frac{4}{2}\\ m=2\end{array}$

The slope of BC will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{3-3}{3-\left(-2\right)}\\ m=\frac{0}{5}\\ m=0\end{array}$

The slope of CD will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-1-3}{5-3}\\ m=\frac{-4}{2}\\ m=-2\end{array}$

The slope of AD will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-1-\left(-1\right)}{-4-5}\\ m=\frac{0}{-9}\\ m=0\end{array}$

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

Hence, ABCD is a trapezoid.