The perpendicular bisector in coordinates

The perpendicular bisector of a segment passes through its midpoint and meets it at a right angle. Find these two ingredients separately, then use a point and gradient to write its equation.

Example

For \(A(1,2)\) and \(B(5,4)\), midpoint \(M=(3,3)\) and gradient of AB is \(\dfrac{4-2}{5-1}=\dfrac12\). The perpendicular gradient is −2, so \(y-3=-2(x-3)\), or \(y=-2x+9\).

A horizontal segment has a vertical perpendicular bisector \(x=x_M\); a vertical segment has a horizontal bisector \(y=y_M\). Avoid taking a reciprocal of zero.