Chapter 3, Problem 22SGR

To determine

To Find: Whether $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$ are perpendicular, parallel or neither and then graph the line to verify the answer.

Answer to Problem 22SGR

The lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$ are perpendicular to each other as shown in Figure 1

Explanation of Solution

Given:

The given vertices is $A\left(-3,9\right),B\left(0,7\right),X\left(4,13\right),Y\left(-5,7\right)$ .

Calculation:

Consider the slope of the line $\overleftrightarrow{AB}$ with $A\left(-3,9\right)$ and $B\left(0,7\right)$ , then the slope is,

  $\begin{array}{c}m=\frac{7-9}{0-\left(-3\right)}\\ =\frac{-2}{3}\end{array}$

The slope of the line $\overleftrightarrow{XY}$ with $X\left(4,13\right)$ and $Y\left(-5,7\right)$ , then the slope of the lie is,

  $\begin{array}{c}m=\frac{7-13}{-5-\left(-14\right)}\\ =\frac{2}{3}\end{array}$

Then, the relation between the slopes is,

  $\begin{array}{c}\frac{-2}{3}\left(\frac{2}{3}\right)=-\frac{4}{9}\\ \ne -1\end{array}$

The production of the slope is not $-1$ so the lines are perpendicular to each other as shown in Figure 1

  

Figure 1