CAT Angle Subtended by a Chord at a Point - Practice Questions & MCQ

Theorem 1 : Equal chords of a circle subtend equal angles at the centre.

Let's see how this is possible.

Let suppose you are given two equal chords AB and CD of a circle with centre O. Consider the figure given below.

And we need to prove if  ∠ AOB = ∠ COD.

In triangles AOB and COD, we have

                           OA = OC         (Radii of a circle)

                          OB = OD          (Radii of a circle)

                          AB = CD          (Given)

Therefore,   ∆ AOB ≅ ∆ COD    (by SSS rule)

This gives  ∠ AOB = ∠ COD     (Corresponding parts of congruent triangles)

Now if two chords of a circle subtend equal angles at the centre, thrn the chords are equal. This is the converse of above theorem.

Theorem 2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

Let a circle with center O in which AB and CD are chords such that ∠ AOB = ∠ COD.

In triangles AOB and COD, we have

                           OA = OC         (Radii of a circle)

                          OB = OD          (Radii of a circle)

                    ∠ AOB = ∠ COD          (Given)

Therefore,   ∆ AOB ≅ ∆ COD    (by SAS rule)

This gives         AB = CD     (Corresponding parts of congruent triangles)