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?

$5\sqrt{3}$ cm

The correct answer is Option (2) → $5\sqrt{3}$ cm

Given: Two tangents are drawn to a circle of radius 5 cm, and the angle between the tangents is $60^\circ$.

Each tangent forms a right-angled triangle with the radius (tangent is perpendicular to radius at the point of contact).

Draw radii to the points of contact. The angle between the radii is $180^\circ - 60^\circ = 120^\circ$ because the tangents meet outside the circle and subtend an angle of $60^\circ$.

So, each triangle formed is an isosceles triangle with two sides of length $r = 5$ cm and the included angle = $120^\circ$.

Use the Cosine Rule to find the length of the tangent (side opposite $120^\circ$):

$l^2 = r^2 + r^2 - 2r^2\cos(120^\circ)$

$l^2 = 25 + 25 - 2 \cdot 25 \cdot \cos(120^\circ)$

$\cos(120^\circ) = -\frac{1}{2}$

$l^2 = 50 + 25 = 75$

$l = \sqrt{75} = 5\sqrt{3}$