Two tangents making an angle of 120 degree with each other are drawn to a circle of radius 6 cm - Maths - Circles

Question

Two tangents making an angle of 120 degree with each other are
drawn to a circle of radius 6 cm then the length of each tangent is
equal to ?

Aakash Sharma · Accepted Answer

Let P be the external points and PA and PB be the two tangents drawn to the circle with centre O.

Given : ∠APB = 120°

Also radius = OA = OB = 6 cm

We know that tangents drawn from an external point are equally inclined to the line joining that point to the centre

$\Rightarrow ∠ APO = ∠ BPO = \frac{∠ APB}{2} = \frac{120 °}{2} = 60 °$

Now In ∆OAP

∠OAP = 90° (∵ Tangent is perpendicular to the radius at the point of contact)

Now in right ∆OAP

∠OPA = 60°

$\Rightarrow \tan 60 ° = \frac{OA}{AP} \Rightarrow \sqrt{3} = \frac{6 cm}{AP} \Rightarrow AP = \frac{6}{\sqrt{3}} cm = \frac{6 \sqrt{3}}{\sqrt{3} \times \sqrt{3}} cm = \frac{6 \sqrt{3}}{3} cm = 2 \sqrt{3} cm$

Also tangents drawn from an external point are equal in length

$\Rightarrow PA = PB = 2 \sqrt{3} cm$

Hence the length of each tangent is $2 \sqrt{3}$ cm.

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Two tangents making an angle of 120 degree with each other are drawn to a circle of radius 6 cm then the length of each tangent is equal to .......?