Question

The equation of the perpendicular bisector of the segment joining $A(-9,2)$ to $B(3,-4)$ is

• A. $y-1=\frac{-1}{2}(x-3)$
• B. $y+1=\frac{-1}{2}(x+3)$
• C. $y+1=2(x+3)$
• D. $y+3=2(x+1)$
• E. $y-1=2(x-3)$

Answer 1

The correct option is C $y+1=2(x+3)$

The slope of the line $\overline{AB}$ is given by:

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-4-2}{3-(-9)}=\frac{-6}{12}=-\frac{1}{2}$

If the slope of the line $\overline{AB}$ is $-\frac{1}{2}$ then the slope of the perpendicular line is $2$.

Now, the midpoint of$\overline{AB}$ is

$(\frac{-9+3}{2},\frac{2-4}{2})=(-\frac{6}{2},-\frac{2}{2})=(-3,-1)$

Since the equation of the line is $y-y_1=m(x-x_1)$ with slope $m$, therefore, the equation of the perpendicular bisector with $m=2$, $x_1=-3$ and $y_1=-1$ is:

$\Rightarrow y-y_1=m(x-x_1)$

$\Rightarrow y-(-1)=2(x-(-3\left)\right)$

$\Rightarrow y+1=2(x+3)$

Hence, the equation of the perpendicular bisector is $y+1=2(x+3)$.