Question

In the adjoining figure, AB is a diameter of a circle with centre O. If chord AM = chord AN, prove that arc BM = arc BN.

Answer 1

Given: AB is a diameter of the circle with centre O. Chord AM = chord AN.

To Prove: arc BM = arc BN.

Proof [Step-wise]:

1. Equal chords subtend equal arcs or equal central angles. 

From AM = AN, we get

arc AM = arc AN

2. Because AB is a diameter, OA and OB are collinear.

So, the arc BA is a semicircle (measure 180°). 

The semicircle BA is partitioned into the two minor arcs AM and BM.

Hence, arc BA = arc AM + arc BM. 

Similarly, arc BA = arc AN + arc BN.

3. Subtract arc AM = arc AN (From step 1) from arc BA in both equations to obtain arc BM = arc BN. 

Explicitly: arc BM = arc BA – arc AM and arc BN = arc BA – arc AN.

So, arc BM = arc BN.