Question

two tangent pq and pr are drawn toa circle with center o from n external point p . prove that angle qpr=2angle oqr

Answer 1

[FIGURE IS IN THE ATTACHMENT]

SOLUTION:

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 helps you