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3. Theorems on angle subtended by the chord at the centre

Angle Subtended by Chord at the Centre

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.)

$\angle POQ = \angle ROS$ .

Proof of the theorem:

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 corresponding parts of it are equal.

Therefore, the equal chords

$PQ$ and

$RS$ subtend equal angles at the centre.

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

$x = 45^{\circ}$ .

Converse of the theorem: If the angles subtended by the chords of a circle at the centre 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.)

$PQ = RS$ .

Proof of the theorem:

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.

Since the triangles are congruent, the corresponding parts of it are equal.

Therefore, the two chords

$PQ$ and

$RS$ are equal.

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.

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3. Theorems on angle subtended by the chord at the centre

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