Question

prove that the angle subtended by an arc at the center is doubled the angle subtended by it at the remaining part of the circle​

Answer 1

Answer:

Given :

An arc PQ of a circle subtending angles POQ at the centre 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)

For the case 3, where PQ is the major arc, equation 4 is replaced by

Reflex angle, ∠POQ=2∠PAQ