GraphicMaths

Visualising maths

Perpendicular bisector of a chord

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

chord bisector

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

chord bisector

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

Interactive example

In this interactive resource, try dragging the red dots around the circumference of the circle:

As you can see the perpendicular bisector always goes through the centre.

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:

chord bisector

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):

chord bisector

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

chord bisector

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.