# If the slope of a line is 17/13, what is the slope of a perpendicular line?

Oct 21, 2015

$- \frac{13}{17}$

#### Explanation:

Suppose a line has equation $y = m x + c$

This is in slope intercept form with slope $m$ and intercept $c$

If we reflect this line in the line $y = x$ then that is equivalent to swapping $x$ and $y$ in the equation, resulting in a line with equation:

$x = m y + c$

If we then reflect that line in the $x$ axis, that is equivalent to replacing $y$ with $- y$, so we get a line with equation:

$x = - m y + c$

Subtract $c$ from both sides to get:

$x - c = - m y$

Divide both sides by $- m$ to get:

$y = - \frac{1}{m} x + \frac{c}{m}$

This is in slope intercept format.

Notice that the geometric result of the two reflections is a rotation through a right angle (Try it yourself with a square of paper with an arrow on one side).

Notice that the effect on the slope is to replace $m$ by $- \frac{1}{m}$.

Any parallel line will just have a different intercept value, so we have shown that a line perpendicular to a line of slope $m$ has slope $- \frac{1}{m}$.