Calculating the Length of the Direct Common Tangent

This problem asks us to find the length of the direct common tangent between two circles with given radii and the distance between their centers. A direct common tangent is a line segment that is tangent to both circles on the same side of the line joining their centers.

Understanding the Given Information

We are provided with the following details:

• Radius of the first circle ($r_1$): 2 cm
• Radius of the second circle ($r_2$): 6 cm
• Distance between the centers of the two circles ($d$): 5 cm

Our goal is to find the length of the direct common tangent ($L$).

Formula for Direct Common Tangent Length

The length of the direct common tangent ($L$) between two circles with radii $r_1$ and $r_2$ (where $r_2 \ge r_1$) and the distance between their centers $d$ is given by the formula:

$$L = \sqrt{d^2 - (r_2 - r_1)^2}$$

This formula is derived using the Pythagorean theorem by constructing a rectangle and a right-angled triangle. Imagine drawing a line parallel to the direct common tangent from the center of the smaller circle to meet the radius of the larger circle perpendicular to the tangent. This forms a right-angled triangle where the hypotenuse is the distance between the centers ($d$), one leg is the difference in radii ($r_2 - r_1$), and the other leg is the length of the direct common tangent ($L$).

Step-by-Step Calculation of Direct Common Tangent

Now, let's substitute the given values into the formula:

Given:

• $r_1 = 2$ cm
• $r_2 = 6$ cm
• $d = 5$ cm

Difference in radii:

$$r_2 - r_1 = 6 \text{ cm} - 2 \text{ cm} = 4 \text{ cm}$$

Using the formula:

$$L = \sqrt{d^2 - (r_2 - r_1)^2}$$

$$L = \sqrt{(5 \text{ cm})^2 - (4 \text{ cm})^2}$$

$$L = \sqrt{25 \text{ cm}^2 - 16 \text{ cm}^2}$$

$$L = \sqrt{9 \text{ cm}^2}$$

$$L = 3 \text{ cm}$$

The length of the direct common tangent is 3 cm.

Summary of the Calculation

Let's summarise the key values and the result:

Revision Table: Direct Common Tangent Formula

It's helpful to remember the formula for the direct common tangent. Here's a quick revision table.

Additional Information: Types of Common Tangents

Besides direct common tangents, there are also transverse common tangents. Understanding the difference is important in geometry problems involving circles.

• Direct Common Tangents: These tangents keep both circles on the same side of the tangent line. The formula used above is for this type. Two direct common tangents can exist unless the circles intersect or one is inside the other.
• Transverse Common Tangents: These tangents pass between the two circles, keeping the circles on opposite sides of the tangent line. The formula for the length of a transverse common tangent ($L_T$) is $L_T = \sqrt{d^2 - (r_1 + r_2)^2}$. Transverse common tangents exist only if the circles are outside each other or touch externally.

In this specific problem, since the distance between centers ($d=5$ cm) is less than the sum of radii ($r_1 + r_2 = 2+6=8$ cm), the circles intersect. When circles intersect, only direct common tangents exist, and transverse common tangents do not. Our calculated length of the direct common tangent is valid.