Question

A and B are points on a circle with centre O. C is a point on the circle such that OC bisects ∠AOB. Prove that OC bisects the arc AB.

Answer 1

Given: A and B are points on a circle with centre O. C is a point on the circle such that OC bisects ∠AOB.

To Prove: OC bisects the arc AB i.e., arc AC = arc CB.

Proof [Step-wise]:

1. OA = OB = OC all are radii of the circle.   ...(Radii of the same circle are equal)

2. OC bisects ∠AOB, so ∠AOC = ∠COB.   ...(Given)

3. Consider triangles ΔAOC and ΔCOB.

OA = OB   ...(Radii)

OC = OC   ...(Common side)

∠AOC = ∠COB   ...(From step 2)

Therefore, ΔAOC ≅ ΔCOB by SAS congruence.

4. From congruence, AC = CB   ...(Corresponding chords are equal)

5. Equal chords subtend equal arcs in the same circle, so arc AC = arc CB.

Thus, OC meets the circle at a point C that divides arc AB into two equal arcs, so OC bisects the arc AB.