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Angle Subtended by Chord at the Centre

Illustration:

Consider a circle with centre $$O$$ and two chords $$PQ$$ and $$RS$$ of equal length.

Join the endpoints of the chords $$PQ$$ and $$RS$$ with the centre $$O$$. Two triangles $$OPQ$$ and $$ORS$$ are obtained with equal sides $$PQ$$ and $$RS$$ (Since the chords are equal).

The other two sides of each triangle are equal as they form the radius of the circle.

Thus, by SSS (Side-Side-Side) rule, the triangles $$OPQ$$ and $$ORS$$ are congruent.

Since the triangles are congruent, the angles subtended by the triangles are also equal.

Based on this congruency, the following theorem is obtained.
Theorem: Equal chords of a circle subtend equal angles at the centre.

Explanation: The theorem states that if the chords $$PQ$$ and $$RS$$ are equal, then the angle subtended by the chords respectively at the centre $$O$$ are equal (i.e.) $$\angle POQ = \angle ROS$$.
Example:
Find the unknown angle $$x$$ in the given figure where the chords $$PQ$$ and $$RS$$ are equal, and $$O$$ is the centre of the circle. Solution:

By the above theorem, the chords $$PQ$$ and $$RS$$ subtend equal angle at $$O$$.

This implies that, $$\angle POQ = \angle ROS$$.

Here, $$\angle ROS = 45^{\circ}$$.

Thus, $$\angle POQ = 45^{\circ}$$.

Therefore, the unknown angle $$x = 45^{\circ}$$.
Illustration:

Following the above theorem, let us find the length of the chords which subtend equal angle at the centre.

Consider a circle with centre $$O$$ and two chords $$PQ$$ and $$RS$$.

Join the endpoints of the chords $$PQ$$ and $$RS$$ with the centre $$O$$.

From the theorem, we have $$\angle POQ = \angle ROS$$. Two triangles $$OPQ$$ and $$ORS$$ are obtained with two sides of each triangle being equal as they form the radius of the circle.

Thus, by SAS (Side-Angle-Side) rule, the triangles $$OPQ$$ and $$ORS$$ are congruent.

Based on this congruency, the following theorem is obtained.
Converse of Theorem: If the angles subtended by two chords at the centre of a circle are equal, then the chords are equal.

Explanation: The theorem states that, if $$PQ$$ and $$RS$$ are two chords subtending equal angle at the centre $$O$$ then the chords $$PQ$$ and $$RS$$ equal (i.e.) $$PQ = RS$$.
Example:
If two parallel chords subtend an equal angle at the centre, then prove that the chords are equal.

Solution:

Given that, the angle subtended by the two chords at the centre are equal.

According to the theorem, if the angles subtended by two chords at the centre of a circle are equal, then the chords are equal.

Therefore, the two parallel chords are equal.

Hence, proved.