Angles Suspended by Chords in a Circle

Theorem: Equal chords of a circle subtend equal angles at the centre of the circle.

In a circle, if we draw two chords of equal lengths then the angles subtended by both the chords at the centre of the circle are equal.

Given: a circle with centre O with two chords of equal length, PQ and RS.

To prove: PQ and RS subtend equal angles at the centre.

i.e ∠POQ = ∠ROS

Proof:

PQ = RS (equal chords given)

OP = OR (Radii of the same circle)

OQ = OS (Radii of the same circle)

Triangles POQ and ROS are congruent: △POQ ≅ △ROS (SSS)

∴ ∠POQ = ∠ROS

Example 7.2.1:

In the diagram below find the value of the chord DC.

Solution:

Chords AB and DC form equal angles at the centre (60°)

We know that If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

Thus, the length of AB and DC are equal.

From the diagram AB = 7

∴ DC = 7