Question

If two tangents are inclined at an angle ${{60}^{\circ }}$, are drawn to a circle of radius $3cm$ then length of each tangent (in CMS) is equal to:

Answer 1

Hint: From the given question we have to find the length of each tangent which is inclined at an angle ${{60}^{\circ }}$. To find the length of the tangent first we have to draw the diagram based on the given information. We have to apply the geometric theory. We know that if two tangents are from the same point then their tangent length will be equal and by joining the centre of the circle and the common point where the tangents intersect there will be two similar triangles. Also, we know that the line from the centre of the circle and the two points of intersection of the tangent will be ${{90}^{\circ }}$i.e., they are perpendicular to each other. by using these theories, we will get the length of the tangents.

Complete step by step solution:
From the questions the tangents are inclined at an angle ${{60}^{\circ }}$.
The assumed figure from the given data is

Let PA and PB be the two tangents to circle with centre O and radius $3cm$.
Firstly, we will find the angles $\angle APO$ and $\angle BPO$.
As we know that these two angles are equal and also, we know that
$\Rightarrow \angle APO=\angle BPO$
$\Rightarrow \angle APB={{60}^{\circ }}$
By this relation we can get the angles,
$\begin{align}
  & \Rightarrow \angle APB=\angle APO+\angle BPO \\
 & \Rightarrow 2\angle APO={{60}^{\circ }} \\
\end{align}$
$\Rightarrow \angle APO=\dfrac{{{60}^{\circ }}}{2}$
$\Rightarrow \angle APO={{30}^{\circ }}=\angle BPO$
also, we know that,
$\Rightarrow OA\bot AP$
$\Rightarrow OB\bot BP$
In a right-angle triangle OAP,
$\Rightarrow \tan {{30}^{\circ }}=\dfrac{3}{PA}$
$\Rightarrow \dfrac{1}{\sqrt{3}}=\dfrac{3}{PA}$
$\Rightarrow PA=3\sqrt{3}$
As we know that the length of the two tangents are equal so,
$\Rightarrow PA=PB=3\sqrt{3}cm$

Note: Students should know the concept of circles and tangents and their properties. A tangent to circle means it will touch or intersect the circle at only one point. The line from the centre of the circle and to the point of intersection of tangents are perpendicular.