# What is the equation of the line that is perpendicular to the line passing through (-5,-6) and (4,-10) at midpoint of the two points?

Equation of the line $18 x - 8 y = 55$

#### Explanation:

From the given two points $\left(- 5 , - 6\right)$ and $\left(4 , - 10\right)$, we need to obtain first the negative reciprocal of the slope m and the midpoint of the points.

Let start with the midpoint $\left({x}_{m} , {y}_{m}\right)$

${x}_{m} = \frac{{x}_{1} + {x}_{2}}{2} = \frac{- 5 + 4}{2} = - \frac{1}{2}$
${y}_{m} = \frac{{y}_{1} + {y}_{2}}{2} = \frac{- 6 + \left(- 10\right)}{2} = - 8$

midpoint $\left({x}_{m} , {y}_{m}\right) = \left(- \frac{1}{2} , - 8\right)$

Negative reciprocal of the slope ${m}_{p} = - \frac{1}{m}$

${m}_{p} = - \frac{1}{m} = \frac{- 1}{\frac{- 10 - - 6}{4 - - 5}} = \frac{- 1}{- \frac{4}{9}} = \frac{9}{4}$

The equation of the line

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

$y - - 8 = \frac{9}{4} \left(x - - \frac{1}{2}\right)$
$y + 8 = \frac{9}{4} \left(x + \frac{1}{2}\right)$
$4 y + 32 = 9 x + \frac{9}{2}$

$8 y + 64 = 18 x + 9$

$18 x - 8 y = 55$

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