Geometry

Circles

Medium

If two tangents inclined at an angle 60° are drawn to a circle of radius 5 cm, then what is the length of each tangent?

5√2 cm

5√3 cm

3 cm

3√3 cm

✅ Correct Option: 2

The circle has radius 5 cm. Two tangents are drawn from an external point, with an angle of 60° between them.

Let O be the center of the circle, P be the external point where the tangents meet, and A and B be the points where the tangents touch the circle.

The line OP bisects the angle between the two tangents.

Since angle APB = 60°:

Angle OPA = $\frac{60°}{2}$

Angle OPA = 30°

Consider triangle OAP:

The tangent is perpendicular to the radius at the point of contact, so angle OAP = 90°.

OA = 5 cm (radius)

PA = length of tangent (to find)

In right triangle OAP:

$\tan(30°) = \frac{OA}{PA}$

$\frac{1}{\sqrt{3}} = \frac{5}{PA}$

$PA = 5\sqrt{3}$ cm

Therefore, the length of each tangent is $5\sqrt{3}$ cm.

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