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.

Using the above, prove the following :

In Figure 4, O is the centre of the circle. If BAO = 30° and BCO = 40°, find the value

of AOC.​

Answer 1

Answer:

Step-by-step explanation:

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