Calculating the Length of Tangents to a Circle with Given Angles

When dealing with circles and tangents, it's important to understand the relationships between the tangents, the radius, and the angles formed. This article will guide you through a detailed step-by-step process to find the length of tangents when two tangents are inclined at a specific angle to a circle.

Understanding the Problem

Consider a circle with a given radius and two tangents drawn from an external point, forming a specific angle. The problem at hand is to determine the length of each tangent. In this example, the radius of the circle is 3 cm, and the angle between the two tangents is 60°.

Step-by-Step Solution

Let's break down the solution step-by-step:

1. Identify the Known Values

Radius of the circle, r: 3 cm Angle between the tangents, θ: 60°

We need to find the length of each tangent, denoted as L.

2. Understand the Geometry

When two tangents are drawn from an external point to a circle, they form an isosceles triangle with the line segment connecting the center of the circle to the external point. The angle at the external point is given, and the angle at the center subtended by the radius to where the tangents touch the circle can be derived.

2.1 Deriving the Central Angle

The angle at the external point is 60°. Since the tangents split this angle equally, the angle subtended by the radius to where the tangents touch the circle is 30°.

3. Calculate the Distance from the Center to the External Point

Using the sine rule, we can express the distance d from the center of the circle to the external point in terms of the radius r and the angle θ/2:

d frac{r}{cos(theta/2)} frac{3}{cos(30^{circ})} frac{3}{frac{sqrt{3}}{2}} frac{6}{sqrt{3}} 2sqrt{3} text{ cm}

4. Apply the Tangent Length Formula

The length of each tangent can be calculated using the formula:

L sqrt{d^2 - r^2}

Substituting the known values:

L sqrt{(2sqrt{3})^2 - 3^2} sqrt{12 - 9} sqrt{3}

Therefore, the length of each tangent is:

boxed{sqrt{3}} text{ cm}

Alternative Method

An alternate way to calculate the length of each tangent involves using trigonometric identities. Here’s how:

Step 1: Identify the Right Triangle Relationships

In right triangle OAP, where O is the center of the circle and P is the external point:

OA: Radius of the circle, 3 cm AP: Length of the tangent to be found Angle OAP: 90° Angle A: 30°

Using the tangent function:

frac{OA}{AP} tan 30^{circ} frac{3}{AP} frac{1}{sqrt{3}}

Solving for AP:

AP 3sqrt{3} text{ cm}

Therefore, the length of each tangent is approximately 5.196 cm.

Conclusion

The methods described above demonstrate how to calculate the length of tangents drawn to a circle from an external point, given the angle between the tangents. Whether using the sine rule or trigonometric identities, the key is to understand the geometric properties of the tangents and the circle.