Chapter 8, Problem 32CCSR

To determine

To prove: The given quadrilateral ABCD is a parallelogram.

Explanation of Solution

Given information:

Consider the points provided for quadrilateral:

A(2,1),B(5,4),C(8,12) and D(5,9)

As, the given points are of a parallelogram then use the points to check whether the opposite sides are parallel or not,

Find the slope of the line AB,

  $\begin{array}{l}{m}_{AB}=\frac{4-1}{5-2}\\ =\frac{3}{3}\\ =1\end{array}$

Find the slope of the line BC,

  $\begin{array}{l}{m}_{BC}=\frac{12-4}{8-5}\\ =\frac{8}{3}\end{array}$

Find the slope of the line CD,

  $\begin{array}{l}{m}_{CD}=\frac{9-12}{5-8}\\ =\frac{-3}{-3}\\ =1\end{array}$

Find the slope of the line DA,

  $\begin{array}{l}{m}_{DA}=\frac{1-9}{5-2}\\ =\frac{8}{3}\end{array}$

Hence, it is cleared that the opposite sides AB and CD have same slope and sides BC and DA have same slope.

Thus, sides AB and CD are parallel and sides BC and DA are parallel. Hence, it is verified that the above points of quadrilateral represents a parallelogram.