If two tangents inclined at an angle of 60 are drawn to circle of radius 13cm , then length of - Maths - Circles

Question

If two tangents inclined at an angle of 60 are drawn to circle
of radius 13cm , then length of each tangent is equal to:

Harleen Birdi · Accepted Answer

Let PA and PB be two tangents to a circle with centre O and radius 13 cm.

We are given ∠APB = 60°

We know that two tangents drawn to a circle from an external point are equally inclined to the segment joining the centre to the point.

∴ ∠APO = ∠BPO = $\frac{1}{2}$ × ∠APB = $\frac{1}{2}$ × 60° = 30°

Also, OA ⊥ AP and OB ⊥ BP (radius ⊥ tangent at point of contact)

In right ΔOAP,

$\tan 30 ° = \frac{13}{PA} \Rightarrow \frac{1}{\sqrt{3}} = \frac{13}{PA} \Rightarrow PA = 13 \sqrt{3} cm$

∴ PA = PB = $13 \sqrt{3} cm$ (Lengths of tangents drawn from an external point to the circle are equal)

If two tangents inclined at an angle of 60 are drawn to circle of radius 13cm , then length of each tangent is equal to: