The length of the direct common tangent of two circles of radius 8 cm and 3 cm is 12 cm. What is the distance (in cm) between the centres of the circles?

Options

A. 15
B. 13
C. 14
D. 17

Answer

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

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

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

### Step-by-Step Solution:

1. **Identify the given values:**
- Radius of the first circle, \(r_1 = 8 \, \text{cm}\)
- Radius of the second circle, \(r_2 = 3 \, \text{cm}\)
- Length of the direct common tangent, \(L = 12 \, \text{cm}\)

2. **Substitute the known values into the formula:**
\[ 12 = \sqrt{d^2 - (8 - 3)^2} \]

3. **Calculate \(r_1 - r_2\):**
\[ r_1 - r_2 = 8 - 3 = 5 \]

4. **Substitute this value back into the equation:**
\[ 12 = \sqrt{d^2 - 5^2} \]

5. **Square both sides to eliminate the square root:**
\[ 12^2 = d^2 - 5^2 \]
\[ 144 = d^2 - 25 \]

6. **Rearrange the equation to solve for \(d^2\):**
\[ d^2 = 144 + 25 \]
\[ d^2 = 169 \]

7. **Take the square root of both sides to find \(d\):**
\[ d = \sqrt{169} \]
\[ d = 13 \, \text{cm} \]

### Conclusion:
The distance between the centers of the circles is \(13 \, \text{cm}\).

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