What is the slope of any line perpendicular to the line passing through (-8,23) and (5,21)?

Jun 14, 2018

See a solution process below:

Explanation:

The formula for find the slope of a line is:

$m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ and $\left(\textcolor{red}{{x}_{2}} , \textcolor{red}{{y}_{2}}\right)$ are two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{21} - \textcolor{b l u e}{23}}{\textcolor{red}{5} - \textcolor{b l u e}{- 8}} = \frac{\textcolor{red}{21} - \textcolor{b l u e}{23}}{\textcolor{red}{5} + \textcolor{b l u e}{8}} = - \frac{2}{13}$

Let's call the slope of a perpendicular line: $\textcolor{b l u e}{{m}_{p}}$

The slope of a line perpendicular to a line with slope $\textcolor{red}{m}$ is the negative inverse, or:

$\textcolor{b l u e}{{m}_{p}} = - \frac{1}{\textcolor{red}{m}}$

Substituting the slope for the line in the problem gives:

$\textcolor{b l u e}{{m}_{p}} = \frac{- 1}{\textcolor{red}{- \frac{2}{13}}} = \frac{1}{\textcolor{red}{\frac{2}{13}}} = \frac{13}{2}$