CBSE Class 9 Answered

Let arc PQ of a circle with centre O, subtends POQ at the centre and PAQ at a point P which lies on the remaining part of the circle.

To prove: POQ = 2 PAB

Proof: There are 3 cases:

(a) Arc PQ is minor

(b) Arc PQ is a semi-circle

(c) Arc PQ is major

In all the three cases, we have:

BOQ = OAQ + AQO ...(1)

(exterior angle is equal to the sum of int. opp. angles.)

Also, in OAQ,

OA = OQ ... (Radii of a circle)

OAQ = OQA ...(2)

From (1) and (2),

BOQ = 2OAQ …..(3)

Similarly,

BOP = 2OAP ……(4)

Adding (3) and (4),

BOP + BOQ = 2(OAP + OAQ)

Or, POQ = 2PAQ ...(5)