Chapter 13.2, Problem 14PSB

To determine

To write: The equation of the perpendicular bisector of $\overleftrightarrow{\text{AB}}$ .

Answer to Problem 14PSB

The equation of the perpendicular bisector of $\overleftrightarrow{\text{AB}}$ is $y=-7x+65$ .

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

Point-slope formula is given by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Find the slope of line $\overleftrightarrow{\text{AB}}$ .

Substitute $2$ for $x_1$ , $16$ for $x_2$ , $3$ for $y_2$ and $1$ for $y_1$ in the equation $m=\frac{y_2-y_1}{x_2-x_1}$ .

  $\begin{array}{c}m=\frac{3-1}{16-2}\\ =\frac{2}{14}\\ =\frac{1}{7}\end{array}$

Find the mid-point of the line $\overleftrightarrow{\text{AB}}$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{2+16}{2},\frac{1+3}{2}\right)\\ =\left(9,2\right)\end{array}$

Line is perpendicular so slope will be $-7$ .

Determine the equation of the perpendicular bisector of $\overleftrightarrow{\text{AB}}$ .

Substitute $9$ for $x_1$ , $2$ for $y_1$ , $-7$ for $m$ in the equation $y-y_1=m\left(x-x_1\right)$ .

  $\begin{array}{c}y-2=-7\left(x-9\right)\\ y-2=-7x+63\\ y=-7x+65\end{array}$

Therefore, the equation of the perpendicular bisector of $\overleftrightarrow{\text{AB}}$ is $y=-7x+65$ .