Question : If two tangents to a circle of radius 3 cm are inclined to each other at an angle of 60°, then the length of each tangent is:
Option 1: $\frac{3 \sqrt{3}}{4} \mathrm{~cm}$
Option 2: $3 \sqrt{3} \mathrm{~cm}$
Option 3: $3 \mathrm{~cm}$
Option 4: $6 \mathrm{~cm}$
Correct Answer: $3 \sqrt{3} \mathrm{~cm}$
Solution :
Let P be an external point, from where two tangents are drawn to the circle and the angle between them is 60°.
Join OA and OP.
OA = 3, is the radius of the circle.
Also, OP is the bisector of $\angle$P.
So, $\angle$APO = $\angle$CPO = 30°
Since tangents at any point of a circle are perpendicular to the radius through the point of contact.
So, OA $\perp$ AP
From $\triangle$OPA, We get,
$\tan 30°=\frac{OA}{AP}$
⇒ $\frac{1}{\sqrt3}=\frac{3}{AP}$
$\therefore AP =3\sqrt3$
Tangents drawn from an external point are equal.
So, AP = CP = $3\sqrt3\ \text{cm}$
Hence, the correct answer is $3\sqrt3\ \text{cm}$.
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