Question

In the adjoining figure, two chords AB and CD of a circle intersect at M. If AB = CD, prove that arc AD = arc СВ.

Answer 1

Given: Two chords AB and CD of a circle intersect at M and AB = CD.

To Prove: arc AD = arc CB.

Proof [Step-wise]:

1. Since AB = CD (given), equal chords subtend equal (minor) arcs in the same circle. 

Hence, minor arc AB = minor arc CD.

2. In the figure, the minor arc AB decomposes as arc AD + arc DB because the minor arc from A to B passes through D, while the minor arc CD decomposes as arc CB + arc BD because the minor arc from C to D passes through B.

3. From step 1, we have arc AB = arc CD. 

So, arc AD + arc DB = arc CB + arc BD.

4. Subtract the common arc BD from both sides to obtain arc AD = arc CB.

Therefore, arc AD = arc CB, as required.