Question

prove that the angle subtended by the arc at the center of the circle is twice the angle subtended by the same arc at the circumference of the circle ​

Answer 1

Answer:

Step-by-step explanation-

prove : ∠POQ=2∠PAQ

To prove 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)

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

Reflex angle, ∠POQ=2∠PAQ