# Perpendicular bisector of a chord

A *chord* is any line drawn across a circle, from one point on the circumference to another:

The *perpendicular bisector* of a chord is the line that cuts the chord in half, at a right angle:

The perpendicular bisector of a chord passes through the centre of the circle.

This theorem is covered in this video on circle theorems:

## Proof

To prove this theorem, we can form a triangle **AOB**, where **O** is the centre of the circle, and **A** and **B** are the points where the chord meets the circumference:

Since **OA** and **OB** are both radii of the circle, they are of equal length. This means that the triangle is an isosceles triangle (see the two radii rule):

The perpendicular bisector of the side **AB** (the chord) is an axis of symmetry of the isosceles triangle:

We know that the axis of symmetry an isosceles triangle passes through the vertex **O**, (which is the centre of the circle).

Therefore the perpendicular bisector of the chord passes through the centre of the circle.