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To solve this problem, we can use the formula for the length of the transverse common tangent between two circles. If the radii of the two circles are \( r_1 \) and \( r_2 \), and the distance between the centers is \( d \), the length of the transverse common tangent \( T \) is given by:

\[
T = \sqrt{d^2 - (r_1 - r_2)^2}
\]

Where:
- \( T \) is the length of the transverse common tangent.
- \( r_1 \) and \( r_2 \) are the radii of the circles.
- \( d \) is the distance between the centers of the two circles.

### Given:
- \( r_1 = 9 \, \text{cm} \) (radius of the larger circle),
- \( r_2 = 6 \, \text{cm} \) (radius of the smaller circle),
- \( T = 20 \, \text{cm} \) (length of the transverse common tangent).

We need to find \( d \), the distance between the centers of the two circles.

Using the formula:

\[
T = \sqrt{d^2 - (r_1 - r_2)^2}
\]

Substitute the given values:

\[
20 = \sqrt{d^2 - (9 - 6)^2}
\]
\[
20 = \sqrt{d^2 - 3^2}
\]
\[
20 = \sqrt{d^2 - 9}
\]

Now, square both sides to eliminate the square root:

\[
400 = d^2 - 9
\]

Solve for \( d^2 \):

\[
d^2 = 400 + 9
\]
\[
d^2 = 409
\]

Take the square root of both sides:

\[
d = \sqrt{409} \approx 20.22 \, \text{cm}
\]

### Conclusion:
The distance between the two centers is approximately \( 20.22 \, \text{cm} \).

---

Do you need further clarification or details? Here are some related questions you can explore:

1. What is the formula for the length of the internal common tangent between two circles?
2. How would the result change if the radii were swapped?
3. What happens if the circles overlap? How does that affect the tangents?
4. Can you derive the formula for the transverse common tangent?
5. What is the significance of the transverse tangent in practical applications?

**Tip:** When working with geometric problems involving circles, it's helpful to sketch the figure to visualize relationships between tangents and centers.

If the radius of two circles be 6 cm and 9 cm and the length of the transverse common tangent be 20 cm, then find the distance between the two centres.

Learn how to find the distance between the centers of two circles when the radii are 6 cm and 9 cm, and the length of the transverse common tangent is 20 cm. This detailed solution uses key geometry formulas and theorems. Suitable for high school students in Grades 9-12.

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