Find the distance between the centres of the two circles, if their radii are 11 cm and 7 cm, and the length of the transverse common tangent is `sqrt(301) cm`.

Answer

To find the distance between the centers of the two circles given their radii and the length of the transverse common tangent, we can use the formula for the length of the transverse common tangent between two circles.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of the first circle (r1) = 11 cm
- Radius of the second circle (r2) = 7 cm
- Length of the transverse common tangent (L) = √301 cm

2. **Use the Formula for the Length of the Transverse Common Tangent**:
The formula for the length of the transverse common tangent is given by:
\[ L = \sqrt{d^2 - (r1 + r2)^2} \]
where \(d\) is the distance between the centers of the two circles.

3. **Substitute the Known Values into the Formula**:
We can substitute the known values into the formula:
\[ \sqrt{301} = \sqrt{d^2 - (11 + 7)^2} \]
Simplifying the radius sum:
\[ 11 + 7 = 18 \]
Thus, we have:
\[ \sqrt{301} = \sqrt{d^2 - 18^2} \]

4. **Square Both Sides to Eliminate the Square Root**:
Squaring both sides gives us:
\[ 301 = d^2 - 18^2 \]

5. **Calculate \(18^2\)**:
\[ 18^2 = 324 \]
Therefore, we can rewrite the equation as:
\[ 301 = d^2 - 324 \]

6. **Rearrange the Equation to Solve for \(d^2\)**:
Adding 324 to both sides:
\[ d^2 = 301 + 324 \]
\[ d^2 = 625 \]

7. **Take the Square Root to Find \(d\)**:
Taking the square root of both sides:
\[ d = \sqrt{625} = 25 \text{ cm} \]

### Final Answer:
The distance between the centers of the two circles is **25 cm**.