# What is the equation of the line that is perpendicular to the line passing through (5,12) and (-2,-23) at midpoint of the two points?

$x + 5 y = - 26$

#### Explanation:

We need the negative reciprocal of the slope $m$ and the midpoint $M \left({x}_{m} , {y}_{m}\right)$

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{- 23 - 12}{- 2 - 5} = \frac{- 35}{- 7} = 5$

The midpoint:

${x}_{m} = \frac{5 + \left(- 2\right)}{2} = \frac{3}{2}$
${y}_{m} = \frac{12 + \left(- 23\right)}{2} = \frac{- 11}{2}$

The equation

$\left(y - {y}_{m}\right) = \left(- \frac{1}{m}\right) \left(x - {x}_{m}\right)$

$\left(y - \frac{- 11}{2}\right) = \left(- \frac{1}{5}\right) \left(x - \frac{3}{2}\right)$
$5 \left(y + \frac{11}{2}\right) = - x + \frac{3}{2}$

$5 \left(2 y + 11\right) = - 2 x + 3$
$10 y + 55 = - 2 x + 3$
$2 x + 10 y = - 52$
$x + 5 y = - 26$

God bless....I hope the explanation is useful.