Perpendicular Bisector Calculator

How it works

M = ((x₁ + x₂)/2, (y₁ + y₂)/2) · m_AB = (y₂ − y₁)/(x₂ − x₁) · m⊥ = −1/m_AB · y − y_M = m⊥(x − x_M) · standard form: (x₂ − x₁)x + (y₂ − y₁)y = (x₂ − x₁)x_M + (y₂ − y₁)y_M

The perpendicular bisector of segment AB passes through its midpoint and meets AB at a right angle, so its slope is the negative reciprocal of AB’s slope. Every point on it is the same distance from A and from B, which is why perpendicular bisectors are used to find circumcenters and the center of a circle through given points. A horizontal segment has a vertical bisector x = constant, and a vertical segment a horizontal one.

Worked example

A(1, 2) and B(5, 6): the midpoint is M(3, 4) and the slope of AB is (6 − 2)/(5 − 1) = 1, so the perpendicular slope is −1. Then y − 4 = −1(x − 3), i.e. y = −x + 7 (standard form x + y = 7). AB is √32 = 4√2 ≈ 5.656854 long.

Sources: OpenStax Algebra and Trigonometry 2e — 4.1 Linear Functions (perpendicular lines) · Wolfram MathWorld — Perpendicular Bisector

Frequently asked questions