Question

Two chords PQ and PR of a circle with centre O are equal. Prove that the centre of the circle lies on the bisector of ∠QPR.

Answer 1

Given: In a circle with centre O, two chords PQ and PR are equal PQ = PR.

To Prove: The centre O lies on the bisector of ∠QPR i.e., OP bisects ∠QPR.

Proof [Step-wise]:

1. Join OP, OQ and OR.

2. OQ = OR radii of the circle PQ = PR.   ...(Given)

3. OP = OP   ...(Common side)

4. In ΔOPQ and ΔOPR,

Three pairs of corresponding sides are equal: 

OP = OP

OQ = OR

PQ = PR

Therefore, ΔOPQ ≅ ΔOPR by SSS congruence.

5. From the congruence,

Corresponding angles at P are equal:

∠QPO = ∠OPR

6. Hence, OP bisects ∠QPR.

Therefore, the centre O lies on the bisector of ∠QPR.