Two tangent segments PA and PB are drawn to a circle with centre O, such that angle APB= 120degree - Maths - Circles

Question

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

Lalit Mehra · Accepted Answer

Given: O is the centre of the circle PA and PB are tangents drawn to a circle and âˆ APB = 120Â° To prove: OP = 2AP Proof: In Î”OAP and Î”OBP, OP = OP (Common) âˆ OAP = âˆ OBP (90Â°) (Radius is perpendicular to the tangent at the point of contact) OA = OB (Radius of the circle) âˆ´ Î”OAP $≅$ Î”OBP (RHS congruence criterion) $\Rightarrow ∠ OAP = ∠ OBP = \frac{120 °}{2} = 60 ° (CPCT)$ In Î”OAP, $\cos ∠ OPA = cos 60° = \frac{AP}{OP} ∴ \frac{1}{2} = \frac{AP}{OP}$ â‡’ OP = 2 AP

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