Question

Find an equation for the perpendicular bisector of the line segment whose endpoints(3, -1) and (-9,5).are

Answer 1

To find the equation you need to find the slope:

1. Find the midpoint coordinates:

$\begin{gathered} \text{midpoint}=(\frac{x_1+x_2}{2},\frac{y_2+y_1}{2}) \\ \\ \text{midpoint}=(\frac{3-9}{2},\frac{-1+5}{2}) \\ \\ \text{midpoint}=(-\frac{6}{2},\frac{4}{2}) \\ \\ \text{midpoint}=(-3,2) \end{gathered}$

The perpendicular bisector passes throuhg point (-3,2)

2. Find the slope (m) of the line segment:

$\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \\ m=\frac{5-(-1)}{-9-3}=\frac{5+1}{-12}=-\frac{6}{12}=-\frac{1}{2} \end{gathered}$

3. Fidn the slope of perpendicular bisector:

Perpendicular lines have negative reciprocal slopes:

The slope of perpendicular bisector is: -1/slope of line segment:

$m=-\frac{1}{-\frac{1}{2}}=2$

4. Find the y-intercept (b) of the perpendicular bisector:

Use the point (-3,2) and the slope 2

$\begin{gathered} y=mx+b \\ 2=2(-3)+b \\ 2=-6+b \\ 2+6=b \\ 8=b \end{gathered}$

Then, the squation of the perpendicular bisector is:

$y=2x+8$