Lesson 2, Topic 2
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Angles Suspended by Chords in a Circle

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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.

Screenshot 2022 06 06 at 11.09.31

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:

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

Screenshot 2022 06 06 at 11.25.26

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

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