Question

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

• A. $\frac{3\sqrt{3}}{2}cm$
• B. $2\sqrt{3}cm$
• C. $3\sqrt{3}cm$
• D. 6 cm

Answer 1

Answer: (C)

$3\sqrt{3}cm$

The tangent to any circle is perpendicular to the radius of the circle at point of contact.

Let two Tangents originate from point A and touch the circle with centre O at point B & C

Now ABO is a right triangle with angle A as 30° and angle B as 90°

Given OB = 3 = Radius of circle.

Now $\frac{OB}{AB}=tan30°$

=>$AB=\frac{OB}{tan30°}$

= $\frac{3}{\left(\frac{1}{\sqrt{3}}\right)}=3\sqrt{3}$

Similarly, it can be shown that $AC=3\sqrt{3}$

As per properties of tangents/ circles, the length of both tangents AB & AC are equal.