Direct Common Tangent Length Calculation

This solution finds the length of the direct common tangent for two circles using the given radii and distance between centers.

Given Information

• Radius of the first circle ($r_1$): $6$ cm
• Radius of the second circle ($r_2$): $18$ cm
• Distance between centers ($d$): $30$ cm

Formula Recall

The formula to calculate the length ($L$) of the direct common tangent between two circles is:

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

Where $d$ is the distance between the centers, and $r_2$ and $r_1$ are the radii of the two circles ($r_2 \ge r_1$).

Calculation Steps

1. Determine the difference in radii:

   $ r_2 - r_1 = 18 \text{ cm} - 6 \text{ cm} = 12 \text{ cm} $

2. Plug the values into the formula:

   $ L = \sqrt{(30 \text{ cm})^2 - (12 \text{ cm})^2} $

3. Calculate the squares:

   $ L = \sqrt{900 \text{ cm}^2 - 144 \text{ cm}^2} $

4. Subtract the values inside the square root:

   $ L = \sqrt{756 \text{ cm}^2} $

5. Simplify the square root:

   Factor $756$ to find perfect squares: $756 = 36 \times 21$.

   $ L = \sqrt{36 \times 21} \text{ cm} $

   $ L = 6\sqrt{21} \text{ cm} $

Conclusion

The length of the direct common tangent is $6\sqrt{21}$ cm.