# What is the equation of the line that is perpendicular to the line passing through (-5,3) and (4,9) at midpoint of the two points?

Apr 7, 2016

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

#### Explanation:

The slope a line that is perpendicular to a given line would be the inverse slope of the given line

$m = \frac{a}{b}$ the perpendicular slope would be $m = - \frac{b}{a}$

The formula for the slope of a line based upon two coordinate points is

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

For the coordinate points $\left(- 5 , 3\right) \mathmr{and} \left(4 , 9\right)$
${x}_{1} = - 5$
${x}_{2} = 4$
${y}_{1} = 3$
${y}_{2} = 9$

$m = \frac{9 - 3}{4 - \left(- 5\right)}$

$m = \frac{6}{9}$

The slope is $m = \frac{6}{9}$
the perpendicular slope would be the reciprocal (-1/m)
$m = - \frac{9}{6}$

To find the midpoint of the line we must use the midpoint formula

$\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$

$\left(\frac{- 5 + 4}{2} , \frac{3 + 9}{2}\right)$

$\left(- \frac{1}{2} , \frac{12}{2}\right)$

$\left(- \frac{1}{2} , 6\right)$

To determine the equation of the line use the point slope form
$\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$

Plug in the midpoint in order to find the new equation.
$\left(- \frac{1}{2} , 6\right)$

$\left(y - 6\right) = - \frac{9}{6} \left(x - \left(- \frac{1}{2}\right)\right)$

$y - 6 = - \frac{9}{6} x - \frac{9}{12}$

$y \cancel{- 6} \cancel{+ 6} = - 1 \frac{1}{2} x - \frac{3}{4} + 3$

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