Question

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at points P, Q and R respectively. Calculate : 

1. ∠QOR
2. ∠QPR;

given that ∠A = 60°.

Answer 1

The incircle touches the sides of the triangle ABC and OP ⊥  BC, OQ ⊥ AC, OR ⊥ AB

i. In quadrilateral AROQ,

∠ORA = 90°, ∠OQA = 90°, ∠A = 60°

∠QOR = 360° – (90° + 90° + 60°)

∠QOR = 360° – 240°

∠QOR = 120° 

ii. Now arc RQ subtends ∠QOR at the centre and ∠QPR at the remaining part of the circle.

∴ `∠QPR = 1/2 ∠QOR`

`=> ∠QPR = 1/2 xx 120^circ`

`=>` ∠QPR = 60°