Question

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

Answer 1

`-13/17`

Explanation:

Suppose a line has equation `y = mx + 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 = my + 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 = -my + c`

Subtract `c` from both sides to get:

`x - c = -my`

Divide both sides by `-m` to get:

`y = -1/m x + 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 `-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 `-1/m`.