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Prove that angle substanded by an arc at the center is double the angel substanded by it any point on the remaining part of the circle(according to class 9)please answer ​

Answer»

Step-by-step explanation:

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)

HOPE you got it....

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

Reflex angle, ∠POQ=2∠PAQ


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