Chapter 10.2, Problem 5QR

To determine

To find: The value of missing coordinate.

Answer to Problem 5QR

The answer is:

  $a=4$

Explanation of Solution

Given information:

The four points of the parallelogram $ABCD$ :

  $\begin{array}{c}A=(0,0),\\ B=(1,3),\\ C=(5,3),\\ D=(a,0)\end{array}$

Calculation:

Know that if $ABCD$ is a parallelogram, then the slopes of the segment lines $AB$ and $CD$ are equal because know that the slopes of the parallel lines are equal.

Know that, slope of the line segment $MN$ , where

  $\begin{array}{c}M=\left(x_1,y_1\right)\\ N=\left(x_2,y_2\right)\end{array}$

Is equal to:

  $\frac{y_2-y_1}{x_2-x_1}$

Hence, in the exercise,

  $\begin{array}{c}A=(0,0),\\ B=(1,3),\\ C=(5,3)\text{, }\\ D=(a,0)\end{array}$

Then $ABCD$ is a parallelogram.

So,

  $\begin{array}{c}\frac{3-0}{1-0}=\frac{0-3}{a-5}\\ \Rightarrow 3=-\frac{3}{a-5}\\ \Rightarrow 3(a-5)=-3\\ \Rightarrow 3a-15=-3\\ \Rightarrow 3a=15-3\\ \Rightarrow a=\frac{12}{3}\\ \Rightarrow a=4\end{array}$

Therefore, the required value of missing coordinate $a=4$ .