SL 3.5—Perpendicular bisectors

The perpendicular bisector of a segment is the line through its midpoint at 90°, and every point on it is equally distant from the two endpoints.

Find the midpoint, calculate the segment's gradient, take the negative reciprocal for the perpendicular gradient, then use point–gradient form. The equal-distance property is often more useful than the equation itself.

For A(1,2) and B(5,4), the midpoint is (3,3) and AB has gradient 1/2, so the bisector has gradient −2: y−3=−2(x−3).

Perpendicular slopes multiply to −1 only when both are finite. Do not use the segment's midpoint with the original gradient.