If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to
a. 3√3/2 cm
b. 6 cm
c. 3 cm
d. 3√3 cm

Solution:

Given, two tangents are drawn to a circle of radius 3 cm, inclined at an angle of 60°

We have to find the length of each tangent.

From the figure,

Let PA and PC be the tangents drawn to a circle

PA and PC inclined at 60°

So, ∠APC = 60°

We know that the tangents through an external point to a circle are equal.

So, PA = PC

In triangle OAP and triangle OCP,

PA = PC

OA = OC = radius of circle

OP = OP = common side

By SSS criterion, triangles OAP and OCP are similar,

We know that the radius of a circle is perpendicular to the tangent at the point of contact.

So, ∠OAP = ∠OCP = 90°

Since OA = OC = radius

∠OAP = ∠OCP

∠APC = ∠OAP + ∠OCP

So, 2∠OAP = 60°

∠OAP = 60°/2

∠OAP = 30°

In triangle OAP,

OAP is a right triangle with A at right angle.

tan 30° = OA/AP

By trigonometric ratio of angles,

tan 30° = 1/√3

So, 1/√3 = 3/AP

AP = 3√3 cm

We know, AP = CP = 3√3 cm

Therefore, the length of each tangent is 3√3 cm.

✦ Try This: If two tangents inclined at an angle 60° are drawn to a circle of radius 5 cm, then length of each tangent is equal to

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10

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NCERT Exemplar Class 10 Maths Exercise 9.1 Problem 9

If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to a. 3√3/2 cm, b. 6 cm, c. 3 cm, d. 3√3 cm

Summary:

If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to 3√3 cm

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