Question

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

Answer 1

Given: From a point P. Outside the circle with centre O, PA and PB are tangents drawn and ∠APB = 120°

And, OP is joined.

Required to prove: OP = 2 AP

Construction: Take mid-point M of OP and join AM, join also OA and OB.

Proof:

In right ∆OAP,

∠OPA = 1/2∠APB = 1/2 (120°) = 60°

∠AOP = 90° – 60° = 30° [Angle sum property]

M is mid-point of hypotenuse OP of ∆OAP [from construction]

So, MO = MA = MP

∠OAM = ∠AOM = 30° and ∠PAM = 90° – 30° = 60°

Thus, ∆AMP is an equilateral triangle

MA = MP = AP

But, M is mid-point of OP

So,

OP = 2 MP = 2 AP

– Hence proved.