Question

If the angle between two tangents drawn from an external point P to a circle of radius a and center O, is 60o, then find the length of OP.

Answer 1

Let AP and BP are two tangents from an external point P on a circle with center O, at points A and B respectively.
Angle with AP and BP, ∠APB = 60° [Given]
In ΔAOP and ΔBOP
AP = BP [Tangents drawn from an external point to a circle are equal]
OP = OP [Common]
OA = OB [Radii of same circle]
ΔAOP ≅ ΔBOP [By Side-Side-Side Criterion]
∠OPA = ∠OPB [Corresponding parts of congruent triangles are equal]
Also,
∠OPA + ∠OPB = ∠APB
⇒ ∠OPA + ∠OPA = 60°
⇒ 2∠OPA = 60°
⇒ ∠OPA = 30°
Also, OA ⏊ AP [Tangent drawn at a point on the circle is perpendicular to the radius through point of contact]
In ΔOAP,

As OA = radius of circle = a]

⇒ OP = 2a

Answer 2

In ΔAOP and ΔBOP
AP = BP [Tangents]
OP = OP [Common]
OA = OB [Radii]
ΔAOP ≅ ΔBOP [By Side-Side-Side Criterion]
∠OPA = ∠OPB [Corresponding parts ]
Also,
∠OPA + ∠OPB = ∠APB
⇒ ∠OPA + ∠OPA = 60°
⇒ 2∠OPA = 60°
⇒ ∠OPA = 30°
Also, OA ⏊ AP [Tangent drawn at a point on the circle is perpendicular to the radius through point of contact]
In ΔOAP,

As OA = radius of circle = a]

⇒ OP = 2a