Question

Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle also in the given figure if pqr is 110 then find the value of pqr​

Answer 1

Given :

An arc PQ of a circle subtending angles POQ at the center O and PAQ at a point A on the remaining part of the circle.

To prove : ∠POQ=2∠PAQ  

To prove this theorem we consider the arc AB in three different situations, minor arc AB, major arc AB and semi-circle AB.

Construction :

Join the line AO extended to B.

Proof :

∠BOQ=∠OAQ+∠AQO       …..(1)

Also, in △ OAQ,

OA=OQ                  [Radii of a circle]

Therefore,

∠OAQ=∠OQA      [Angles opposite to equal sides are equal]

∠BOQ=2∠OAQ              ….(2)

Similarly, BOP=2∠OAP             …..(3)

Adding 2 & 3, we get,

∠BOP+∠BOQ=2(∠OAP+∠OAQ)

∠POQ=2∠PAQ               …..(4)

Answer 2

HOPE IT HELPS YOU...........