Calculating the Lengths of Tangent Segments Between Two Circles

In this guide, we will explore how to calculate the lengths of both external tangent and internal tangent segments between two circles given their radii and the distance between their centers. This is a fundamental problem in geometric constructions and has practical applications in various fields, including engineering and design.

Given Parameters and Problem Setup

Consider two circles with radii 2 and 11, and the distance between their centers is 15. We are asked to find the lengths of the external tangent segment and the internal tangent segment.

Calculating the External Tangent Segment

To find the length of the external tangent segment, we use the geometric properties and the Pythagorean theorem. Here are the detailed steps:

Using the Parallelogram Property

First, we draw a line AC parallel to the tangent segment ED. Since AEDC is a parallelogram, AC DE is the required length. To find AC, consider the right triangle ABC:

Using the Pythagorean theorem in the right triangle ABC, we have:

AC2 AB2 - BC2

Here, AB 15 and CD 2. Therefore:

AC2 152 - 11 - 22 225 - 121 144

Thus, AC √144 12 units.

Calculating the Internal Tangent Segment

For the internal tangent segment, we need to consider the similar triangles BCD and EDA. Using the properties of similar triangles, we get:

BC/AD BE/AE CE/ED

Solving for AE and DE, we find:

AE 30/13 units

Note that BE 165/13 units.

Then, using the Pythagorean theorem:

DE2 AE2 - AD2 (30/13)2 - 22 900/169 - 4/1 (900 - 676)/169 224/169

Thus, DE √(224/169) ≈ 1.151 units.

Similarly, in the right triangle BCE for the internal segment:

CE2 BE2 - BC2 (165/13)2 - 112 27225/169 - 121 (27225 - 20449)/169 6776/169

Thus, CE √(6776/169) ≈ 6.332 units.

The length of the internal tangent segment is CD CE ED ≈ 6.332 1.151 ≈ 7.483 units.

General Formulas and Special Cases

In a more general form, the length of the external tangent segment (Texternal) is:

Texternal √(D2 - (Ra - Rb)2)

And the internal tangent segment (Tinternal) is:

Tinternal √(D2 - (Ra Rb)2)

A special case occurs when D Ra Rb:

Texternal 2√(Ra * Rb)

Tinternal 0

Conclusion

The lengths of the external and internal tangent segments can be accurately calculated using the methods and formulas provided. These calculations are essential for a wide range of applications, from solving geometric problems to practical engineering designs.

Do you have any questions or need further clarification on these calculations? Feel free to reach out!