Perpendicular Line Equation

Idea

A line perpendicular to another line has a slope that is the negative reciprocal of the original slope. If one slope is (m), the perpendicular slope is (-\frac{1}{m}) (when (m \neq 0)). If the given line is horizontal, the perpendicular line is vertical; if the given line is vertical, the perpendicular line is horizontal.

Method

1. Find the slope of the given line. If the line is already in slope-intercept form (y=mx+b), read off (m). If not, rewrite it or use two points.
2. Take the negative reciprocal of that slope.
3. Use the point the new line must pass through, if one is given.
4. Write the line equation with point-slope form:
   [y-y_1=m(x-x_1)] Then simplify to the requested form.

Special cases

• If the original slope is (0), the perpendicular line is vertical, so its equation looks like (x=a).
• If the original line is vertical, the perpendicular line is horizontal, so its equation looks like (y=b).

Check

Confirm that the two slopes multiply to (-1) when both lines have defined slopes. For vertical and horizontal lines, check that one equation is of the form (x=) constant and the other is of the form (y=) constant.