Perpendicular bisector of a chord

By Martin McBride, 2020-08-16
Tags: circle chord perpendicular bisector
Categories: gcse geometry


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:

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.

See also



Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

Popular tags

adder adjacency matrix alu and gate angle area argand diagram binary maths cartesian equation chain rule chord circle cofactor combinations complex polygon complex power complex root cosh cosine cosine rule cpu cube decagon demorgans law derivative determinant diagonal directrix dodecagon ellipse equilateral triangle eulers formula exponent exponential exterior angle first principles flip-flop focus gabriels horn gradient graph hendecagon heptagon hexagon horizontal hyperbola hyperbolic function infinity integration by substitution interior angle inverse hyperbolic function inverse matrix irregular polygon isosceles trapezium isosceles triangle kite koch curve l system locus maclaurin series major axis matrix matrix algebra minor axis nand gate newton raphson method nonagon nor gate normal not gate octagon or gate parabola parallelogram parametric equation pentagon perimeter permutations polar coordinates polynomial power product rule pythagoras proof quadrilateral radians radius rectangle regular polygon rhombus root set set-reset flip-flop sine sine rule sinh sloping lines solving equations solving triangles square standard curves star polygon straight line graphs surface of revolution symmetry tangent tanh transformations trapezium triangle turtle graphics vertical volume of revolution xnor gate xor gate