# How do I find the equation of the perpendicular bisector of the line segment whose endpoints are (-4, 8) and (-6, -2) using the Midpoint Formula?

Nov 14, 2015

$y = - \frac{1}{5} x + 2$

#### Explanation:

First, you must find the midpoint of the segment, the formula for which is $\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$. This gives $\left(- 5 , 3\right)$ as the midpoint. This is the point at which the segment will be bisected.

Next, since we are finding a perpendicular bisector, we must determine what slope is perpendicular to that of the existing segment. To determine the segment's slope, we use the slope formula $\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$, which gives us a slope of $5$.

Perpendicular lines have opposite and reciprocal slopes. The opposite reciprocal of $5$ is $- \frac{1}{5}$.

We now know that the perpendicular travels through the point $\left(- 5 , 3\right)$ and has a slope of $- \frac{1}{5}$.

Solve for the unknown $b$ in $y = m x + b$.

$3 = - \frac{1}{5} \left(- 5\right) + b \implies 3 = 1 + b \implies 2 = b$

Therefore, the equation of the perpendicular bisector is $\textcolor{b l u e}{y = - \frac{1}{5} x + 2}$.