Question

in the following figure. OP is equal to diameter of the circle .Prove that ABP is an equilateral triangle.

Answer 1

Step-by-step explanation:

PA and PB are the tangents to the circle.

∴ OA ⊥ PA

⇒ ∠OAP = 90°

In ΔOPA,

 sin ∠OPA = OA OP  =  r 2r   [Given OP is the diameter of the circle]

⇒ sin ∠OPA = 1 2 =  sin  30 ⁰

⇒ ∠OPA = 30°

Simlarly, it can be proved that ∠OPB = 30°.

Now, ∠APB = ∠OPA + ∠OPB = 30° + 30° = 60°

In ΔPAB,

PA = PB         [lengths of tangents drawn from an external point to a circle are equal]

⇒∠PAB = ∠PBA ............(1)   [Equal sides have equal angles opposite to them]

∠PAB + ∠PBA + ∠APB = 180°    [Angle sum property]

⇒∠PAB + ∠PAB = 180° – 60° = 120°  [Using (1)]

⇒2∠PAB = 120°

⇒∠PAB = 60°    .............(2)

From (1) and (2)

∠PAB = ∠PBA = ∠APB = 60°

∴ ΔPAB is an equilateral triangle.