Question

If two tangents inclined at an angle of 60 degrees are drawn to a circle of radius 4 cm then the length of each tangent is equal to:-
A. $2\sqrt 3 cm$
B. $8cm$
C. $4cm$
D. $4\sqrt 3 cm$

Answer 1

Hint: In order to solve this problem you need to draw the diagram of tangents and should know that the line from center bisects the angle between two tangents from a point. Then apply trigonometry to get the answer.

Complete Step-by-Step solution:
The diagram for this problem can be drawn as:-

We know that AO and CO is the radius and with tangent radius make 90 degrees of angle.
So, angle ABO = angle BCO = 90 Degrees.
AO = OC = Radius of the circle.
AB = BC (Since tangents from one point on the same circle are equal)
First we consider triangle ABO,
The angle ABO is 30 degrees since the line from the center of the circle bisects the angle between two tangents from a point.
So we do,
$
  {\text{tan30 = }}\dfrac{{\text{1}}}{{\sqrt {\text{3}} }}{\text{ = }}\dfrac{{{\text{AO}}}}{{{\text{AB}}}}{\text{ = }}\dfrac{{\text{4}}}{{{\text{AB}}}} \\
  {\text{AB = 4}}\sqrt {\text{3}} \,\,\,\,\,\,\, \\
$
Hence AB =${\text{4}}\sqrt {\text{3}} $ cm.
So, the lengths of tangents = AB = BC = ${\text{4}}\sqrt {\text{3}} $cm.
Hence the correct option is B.

Note: In these types of problems of circles you need to use the properties of tangent. Here we have used the properties tangent that the angle between radius and tangent is 90 degrees, tangents from a single point on the circle are of equal length and we have applied trigonometric formulas in triangles as well to get the solution to this problem.