Chapter 6.3, Problem 24PPS

To determine

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

Answer to Problem 24PPS

ABCD is a parallelogram.

Explanation of Solution

Given:

The given vertices are:

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

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:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{5-4}{4+3}\\ m=\frac{1}{7}\end{array}$

The slope of line segment BC will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-1-5}{5-4}\\ m=\frac{-6}{1}\\ m=-6\end{array}$

The slope of line segment CD will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-2-\left(-1\right)}{-2-5}\\ m=\frac{-1}{-7}\\ m=\frac{1}{7}\end{array}$

The slope of line segment AD will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-2-4}{-2-\left(-3\right)}\\ m=\frac{-6}{1}\\ m=-6\end{array}$

The slope of line AB is equal to the slope of line CD and the slope of line BC is equal to the slope of AD.

Therefore, by the definition of a parallelogram, ABCD is a parallelogram.