Chapter 6.3, Problem 6CYU

To determine

The graph of each quadrilateral with given vertices and whether the figure forms a parallelogram or not.

Answer to Problem 6CYU

ABCD is not a parallelogram.

Explanation of Solution

Given:

The given vertices are:

  $A\left(-2,4\right),B\left(5,4\right),C\left(8,-1\right),D\left(-1,-1\right).$

Calculation:

Parallelogram has both pairs of its opposite sides as parallel sides,

The quadrilateral with given vertices will be:

The slope of line segment AB will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{4-4}{5-\left(-2\right)}\\ m=\frac{0}{7}\\ m=0\end{array}$

The slope of line segment BC will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-1-4}{8-5}\\ m=\frac{-5}{3}\\ m=-\frac{5}{3}\end{array}$

The slope of line segment CD will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-1-\left(-1\right)}{-1-8}\\ m=\frac{0}{-9}\\ m=0\end{array}$

The slope of line segment AD will be:

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

The slope of line BC is not equal to the slope of line AD.

Therefore, by the definition of a parallelogram, ABCD is not a parallelogram.