The distance between the centres of two circles of radius 9 cm and 6 cm is 17 cm. What is the length (in cm) of the transverse common tangent?

Options

A. 9
B. 11
C. 7
D. 8

Answer

To find the length of the transverse common tangent between two circles with given radii and the distance between their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of Circle 1 (r1) = 9 cm
- Radius of Circle 2 (r2) = 6 cm
- Distance between the centers (d) = 17 cm

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

3. **Calculate the Sum of the Radii**:
\[ r1 + r2 = 9 + 6 = 15 \text{ cm} \]

4. **Square the Distance and the Sum of the Radii**:
- Calculate \(d^2\):
\[ d^2 = 17^2 = 289 \]
- Calculate \((r1 + r2)^2\):
\[ (r1 + r2)^2 = 15^2 = 225 \]

5. **Substitute the Values into the Formula**:
\[ L = \sqrt{289 - 225} \]

6. **Perform the Subtraction**:
\[ 289 - 225 = 64 \]

7. **Take the Square Root**:
\[ L = \sqrt{64} = 8 \text{ cm} \]

### Final Answer:
The length of the transverse common tangent is **8 cm**.

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