Question

The angle subtended by an arc at centre is double the angle subtended by it at any point on the remaining part of the circle​

Answer 1

Given:

A circle with centre O. Arc PQ of the circle subtends angles POQ at the centre O and <PAQ at the point A remaining part of the circle

To prove:

<POQ=2<PAQ

Construction:

Join AO and extend it to B

Proof:

Now in all three cases;

<BOQ= <OAQ+<AQO--------(i) (as an exterior angle of the triangle is equal to the sum of two interior opposite angles)

also in triangle OAQ

OA=OQ( radii of circle)

Therefore,

<OAQ=OQA (angle opposite to equal sides are equal)

<BOQ= 2<OAQ --------(ii)

similarly, <BOP=2<OAP ---------(iii)

adding ii and iii, we get

<BOP+<BOQ= 2 (OAP+OAQ)

<POQ=2<PAQ------(iv)

reflex angle POQ=2<PAQ

hence proved