Question

two tangents PA and PB are drawn to the circle with Centre O, such that angle APB=60 degree. prove that OP=2AP

Answer 1

THIS IS WITH ANGLE 120 SAME STEPS AS FOLLOWS ONLY ANGLE 60

In ΔOAP and ΔOBP,

OP = OP    (Common)

∠OAP = ∠OBP  (90°)    (Radius is perpendicular to the tangent at the point of contact)

OA = OB  (Radius of the circle)

∴ ΔOAP ≅ ΔOBP  (RHS congruence criterion)

⇒ ∠OAP = ∠OBP = 120°/2 = 60°

In ΔOAP,

cos ∠OPA = 60° = AP/OP

∴ 1/2 = AP/OP

⇒ OP = 2 AP