Question

Q15) Prove that the angles subtended by an arc at the centre is double the angle
subtended by it at any point on the remaining part of the circle.
Also find the value of x
At
G
OR
A circular park of radius 30m is situated in a colony. Three girls Neha , Sneha, P
are sitting at equal distance on its boundary each having a toy telephone in her
hands to talk each other. Find the length of the string of each phone​

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     Step(1)

Also, in △ OAQ,

OA=OQ                  [Radii of a circle]

Therefore,

∠OAQ=∠OQA      [Angles opposite to equal sides are equal]

∠BOQ=2∠OAQ                Step (2)

Similarly, BOP=2∠OAP    Step (3)

Adding 2 & 3, we get,

∠BOP+∠BOQ=2(∠OAP+∠OAQ)

∠POQ=2∠PAQ                 Step (4)

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

Reflex angle, ∠POQ=2∠PAQ

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