Answer
$y=-\frac{4}{9}x-\frac{13}{9}$
Work Step by Step
We are given the points $A(3,8)$ and $B(-5,-10)$.
a) $\bf{Step\text{ }1}$
Find the midpoint of the line segment using $A(x_1,y_1)=(3,8)$, $B(x_2,y_2)=(-5,-10)$:
$$\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)=\left(\dfrac{3+(-5)}{2},\dfrac{8+(-10)}{2}\right)=(-1,-1)$$
b) $\bf{Step\text{ }2}$
Calculate the slope $m$ of $\overline{AB}$:
$$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-10-8}{-5-3}=\dfrac{9}{4}.$$
$\bf{Step\text{ }3}$
Find the slope $m_1$ of the perpendicular bisector:
$$\begin{align*}
m_1&=-1\\
m_1&=-\dfrac{1}{m_1}=-\dfrac{1}{\frac{9}{4}}=-\dfrac{4}{9}.
\end{align*}$$
$\bf{Step\text{ }4}$
Use point-slope form:
$$\begin{align*}
y-(-3)&=m_1(x-1)\\
y-(-1)&=-\dfrac{4}{9}(x-(-1))\\
y&=-\dfrac{4}{9}x-\dfrac{13}{9}.
\end{align*}$$
The equation of the perpendicular bisector is
$$y=-\dfrac{4}{9}x-\dfrac{13}{9}.$$
