Question

Two tangents are drawn to the circle at an angle of ${60}^o$ what is the length of each tangent if chord toching the point of contacts of both the tangents is $6$ cm?

• A. $3\sqrt{3}$ cm
• B. $6$ cm
• C. $6\sqrt{3}$ cm
• D. $3$ cm

Answer 1

The correct option is B $6$ cm


Concept: Two tangents drawn from the point ouside the circle are equal in length.
In a triangle, the angles opposite to equal sides are equal.

In $△ABC$
$AC=BC$$\text{(two tangents from one point}\text{to a circle are equal in length)}$
So, $\angle A=\angle B\text{(angle opposite to equal sides are equal)}$
Now,
$\angle A+\angle B+\angle C={180}^o\text{(angle sum property of triangle)}$
$\angle A+\angle A+{60}^o={180}^o$
$2\angle A={120}^o$
$\angle A={60}^o$
So, $\angle A=\angle B={60}^o$

As all angles of $△ABC$ are ${60}^o$ so triangle is equilateral so all sides of the triangle are also equal to each other.
So, $AB=BC=AC=6$ cm
Length of tangents is $6$ cm

Hence, option $\left(b\right)$ correct.