What is the equation of the line that is perpendicular to the line passing through (-5,3) and (-2,9) at midpoint of the two points?

Feb 2, 2018

$y = - \frac{1}{2} x + \frac{17}{4}$

Explanation:

$\text{we require to find the slope m and the midpoint of the}$
$\text{line passing through the given coordinate points}$

$\text{to find m use the "color(blue)"gradient formula}$

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)

$\text{let "(x_1,y_1)=(-5,3)" and } \left({x}_{2} , {y}_{2}\right) = \left(- 2 , 9\right)$

$\Rightarrow m = \frac{9 - 3}{- 2 - \left(- 5\right)} = \frac{6}{3} = 2$

$\text{the slope of a line perpendicular to this is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m=-1/2

$\text{the midpoint is the average of the coordinate of the}$
$\text{given points}$

$\Rightarrow M = \left[\frac{1}{2} \left(- 5 - 2\right) , \frac{1}{2} \left(3 + 9\right)\right] = \left(- \frac{7}{2} , 6\right)$

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

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\Rightarrow y = - \frac{1}{2} x + b \leftarrow \textcolor{b l u e}{\text{is partial equation}}$

$\text{to find b substitute the coordinates of the midpoint}$
$\text{into the partial equation}$

$6 = \frac{7}{4} + b \Rightarrow b = \frac{17}{4}$

$\Rightarrow y = - \frac{1}{2} x + \frac{17}{4} \leftarrow \textcolor{red}{\text{perpendicular line}}$