Q15 of 47 Page 10

Two tangent segments PA and PB are drawn to a circle with centre O such that ∠APB=120°. Prove that OP = 2 AP.

Given: ∠APB=120°


To find: OP = 2 AP


Theorem Used:


1.) The length of two tangents drawn from an external point are equal.


2.) Tangent to a circle at a point is perpendicular to the radius through the point of contact.


Explanation:



P is the external point, PA and PB are the tangents.


From the theorem stated PA = PB.


In ΔOAP and ΔOPB


PA = PB


OA = OB (Radii of same circle)


OP = OP (common)


By SSS criterion,


ΔOAP~ΔOPB


∠OPA = ∠OPB = 60o.


AP is tangent to radius OA.


By theorem (2) stated above,


∠OAP = 90o




So, OP = 2 AP


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