Question

Write an equation of the perpendicular bisector of the segment with the endpoints G(3,7) and H(-1,-5).

Answer 1

ANSWER

$y=-\frac{1}{3}x+\frac{4}{3}$

EXPLANATION

We want to find the equation of the perpendicular bisector of the segment with the endpoints G(3,7) and H(-1,-5).

Since the line is a bisector, it means that it passes through the midpoint of G and H.

Also, since it is perpendicular to the line with endpoints G and H, it means that the slope is the negative inverse of the slope of the line between the two points.

First, find the midpoint of the two points G and H:

$\begin{gathered} M=(\frac{x1+x2}{2},\frac{y1+y2}{2}) \\ M=(\frac{3+(-1)}{2},\frac{7+(-5)}{2}) \\ M=(\frac{2}{2},\frac{2}{2}) \\ M=(1,1) \end{gathered}$

Next, find the slope of the line between points G and H:

$\begin{gathered} m=\frac{y2-y1}{x2-x1} \\ m=\frac{-5-7}{-1-3}=\frac{-12}{-4} \\ m=3 \end{gathered}$

Now, find the negative inverse:

$m_2=-\frac{1}{m_1}=-\frac{1}{3}$

Find the equation of the line using the point-slope method:

$\begin{gathered} y-y1=m(x-x1) \\ y-1=-\frac{1}{3}(x-1) \\ y-1=-\frac{1}{3}x+\frac{1}{3} \\ y=-\frac{1}{3}x+\frac{1}{3}+1 \\ y=-\frac{1}{3}x+\frac{4}{3} \end{gathered}$

That is the equation of the line.