# How do you find the equation for the perpendicular bisector of the segment with endpoints (-1,-3) and (7,1)?

May 18, 2018

$y = - 2 x + 5$

#### Explanation:

$\text{the perpendicular bisector bisects the line segment at}$
$\text{right angles}$

$\text{we require to find the midpoint of the segment and }$
$\text{the slope m}$

$\text{the midpoint of any endpoints say "(x_1,y_1)" and "(x_2,y_2)" is}$

•color(white)(x)[1/2(x_1+x_2),1/2(y_1+y_2)]

$\text{midpoint } = \left[\frac{1}{2} \left(- 1 + 7\right) , \frac{1}{2} \left(- 3 + 1\right)\right]$

$\textcolor{w h i t e}{\text{midpoint }} = \left[\frac{1}{2} \left(6\right) , \frac{1}{2} \left(- 2\right)\right] = \left(3 , - 1\right)$

$\text{calculate slope m using the "color(blue)"gradient formula}$

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)

$\text{let "(x_1,y_1)=-1,-3)" and } \left({x}_{2} , {y}_{2}\right) = \left(7 , 1\right)$

$\Rightarrow m = \frac{1 - \left(- 3\right)}{7 - \left(- 1\right)} = \frac{4}{8} = \frac{1}{2}$

$\text{given a line with slope m then the slope of a line}$
$\text{perpendicular to it is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{\frac{1}{2}} = - 2$

$\text{the equation of a line in "color(blue)"point-slope form}$ is.

•color(white)(x)y-y_1=m(x-x_1)

$\text{where m is the slope and "(x_1,y_1)" a point on the line}$

$\text{using "m=-2" and "(x_1,y_1)=(-1,-3)" then}$

$\Rightarrow y + 1 = - 2 \left(x - 3\right) \leftarrow \textcolor{red}{\text{in point-slope form}}$

$\Rightarrow y + 1 = - 2 x + 6$

$\Rightarrow y = - 2 x + 5 \leftarrow \textcolor{red}{\text{in slope-intercept form}}$