Question

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

Answer 1

The complete question is,

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

Hence it is proved that OP = 2 AP.

Given,

O is the center of the circle.

PA and PB are the tangents drawn to the circle.

∠ APB = 120°

To prove:

OP = 2 AP

Proof:

In Δ OAP and Δ OBP

OP = OP              (common side)

∠ OAP = ∠ OBP = 90°       (given)

OA = OB           (radius of the same circle)

∴ Δ OAP ≅ Δ OBP         (SAS theorem criteria)

⇒ ∠ OAP = ∠ OBP = 120°/2 = 60°

In Δ OAP

Cos OAP = AP/OP

Cos 60° = AP/OP

1/2 = AP/OP

∴ OP = 2AP.