Question

Two tangents PQ &PR are drawn to a circle with center O from an external point P. prove that angle QPR =angle OQR

Answer 1

Answer:

proved below

Step-by-step explanation:

Given that: PQ and PR are two tangents drawn to a circle with centre O from an external point P

To Prove: ∠QPR=2∠OQR

Construction: Join QR, OQ and OR

Proof: we know that lengths of a tangent drawn from an external point to a circle are equal.

PQ= PR

∆PQR is an isosceles triangle.

∠PQR= ∠PRQ

In ∆ PQR

∠PQR+ ∠PRQ+∠QPR= 180°

∠PQR+ ∠PQR+∠QPR= 180°

2∠PQR= 180°-∠QPR

∠PQR=1/2 (180°-∠QPR)

∠PQR= 90°-1/2∠QPR

1/2∠QPR=90°-∠PQR……….  (1)

Since, OQ Perpendicular PQ

∠OQP= 90°

∠OQR+ ∠PQR=90°

∠OQR =90°- ∠PQR……..(2)

∠OQR =1/2∠QPR

2∠OQR =∠QPR

∠QPR=2∠OQR

Hope this will help you....