Question

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

Answer 1

Given that

Angle between two tangent is 60°

angle APB = 60°

Now,

In ∆OPA and ∆ OPB

angle OAP = angle OPB ( both 90°)

OP = OP ( common )

OA = OB ( both radius )

∆ OPA = ∆ OPB ( RHS congruency )

angle OPA = angle OPB ( CPCT )

so, we can write ,

angle OPA = angle OPB = 1/2 angle APB

so, angle OPA = 1/2 × 60° = 30°

Now , In ∆ OPA

sin P = OA/OP

sin 30° = r/OP

1/2 = r/OP

OP = 2r