Question

prove that angle subtended by chord at centre is double the angle subtended by it at boundary of circle​

Answer 1

Answer:

Given :

* 76

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 : ZP OQ = 2ZP AQ AQ3D

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:

ZBOQ = 20AQ + ZIQO ..(1)

Also, in A OAQ,

OA=OQ

Therefore,

[Radii of a circle]

ZOAQ = ZOQA [Angles opposite to equal

sides are equal]

ZBOQ=220AQ .(2)

Similarly, BOP = 220AP .(3)

Adding 2 & 3, we get,

ZBOP + ZBOQ = 2(2 OAP + ZOAQ)

ZPOQ=2ZP AQ .(4)

For the case 3, where PQ is the major arc, equation 4 is replaced by Reflex angle, ZPOQ=2ZP AQ