Question

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​

Answer 1

$\huge {\bold {Question}}$

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 .

$\huge {\bold {\blue {s0lUtion}}}$

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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

Hence, proved.

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$\huge {\bold {\pink {Hope \: it \: helps}}}$

Answer 2

Answer:

Here is the answer

Step-by-step explanation:

Hope it helped u