Question

prove that the angle subtended by an arc at the center is double the angle subtended by it at any point on remaining part of the circle

Answer 1

Answer:

Step-by-step explanation:

An arc PQ of a circle is given, 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 minor arc AB, major arc AB and semi-circle AB.

We

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)

, where PQ is the major arc, equation 4 is replaced by

Reflex angle, ∠POQ=2∠PAQ

Answer 2

see the attachment for answer