# What is the slope of any line perpendicular to the line passing through (15,-22) and (12,-15)?

Oct 2, 2016

$m = \frac{3}{7}$

#### Explanation:

Given 2 perpendicular lines with slopes ${m}_{1} \text{ and } {m}_{2}$ then

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{m}_{1} \times {m}_{2} = - 1} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

We require to calculate ${m}_{1}$ using the $\textcolor{b l u e}{\text{gradient formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

The 2 points here are (15 ,-22) and (12 ,-15)

$\Rightarrow {m}_{1} = \frac{- 15 - \left(- 22\right)}{12 - 15} = \frac{7}{- 3} = - \frac{7}{3}$

Thus $- \frac{7}{3} \times {m}_{2} = - 1$

$\Rightarrow {m}_{2} = \frac{- 1}{- \frac{7}{3}} = \frac{3}{7}$

Hence the slope of any line perpendicular to the line passing through the 2 given points is $m = \frac{3}{7}$