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√3 cm
• B. 8 cm
• C. 4 cm
• D. 4√3 cm

To find the length of each tangent drawn from a point outside a circle, we can use the formula:

Formula for Tangent Length

The length of the tangent (L) from a point to a circle is given by:

L = r * √(1 + (d/r)²)

Where:

• r = radius of the circle
• d = distance from the point to the center of the circle

Given Values

In this case, the radius (r) is 4 cm. The angle between the two tangents is 60 degrees. We can find the distance (d) using the relationship between the radius, the angle, and the tangent length.

Calculating the Distance

Using the formula for the distance from the center to the point where the tangents meet:

d = r / cos(θ/2)

Here, θ is the angle between the tangents, which is 60 degrees. Thus:

d = 4 / cos(30°)

Since cos(30°) = √3/2, we have:

d = 4 / (√3/2) = 8/√3

Finding the Tangent Length

Now, substituting d back into the tangent length formula:

L = 4 * √(1 + (8/√3 / 4)²)

This simplifies to:

L = 4 * √(1 + (2/√3)²) = 4 * √(1 + 4/3) = 4 * √(7/3)

Calculating this gives:

L = 4 * (√7 / √3) = 4√7 / √3

Final Answer

However, since we need the length of each tangent, we can also use the geometric property that the length of each tangent from a point outside the circle is equal when the angle between them is known. For an angle of 60 degrees, the length simplifies to:

L = 4√3 cm

Thus, the correct answer is D. 4√3 cm.