The distance between the centres of two circles of radius 2.5 cm each is 13 cm. The length (in cm ) of a transverse common tangent is

Options

A. 10
B. 12
C. 8
D. 6

Answer

To find the length of the transverse common tangent between two circles, we can use the formula:

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

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

### Step-by-Step Solution:

1. **Identify the given values**:
- The distance between the centers of the circles, \(d = 13 \, \text{cm}\).
- The radius of each circle, \(r_1 = r_2 = 2.5 \, \text{cm}\).

2. **Calculate the sum of the radii**:
\[ r_1 + r_2 = 2.5 + 2.5 = 5 \, \text{cm} \]

3. **Substitute the values into the formula**:
\[ L = \sqrt{d^2 - (r_1 + r_2)^2} \]
\[ L = \sqrt{13^2 - 5^2} \]

4. **Calculate \(d^2\) and \((r_1 + r_2)^2\)**:
\[ d^2 = 13^2 = 169 \]
\[ (r_1 + r_2)^2 = 5^2 = 25 \]

5. **Subtract the squares**:
\[ L = \sqrt{169 - 25} = \sqrt{144} \]

6. **Find the square root**:
\[ L = 12 \, \text{cm} \]

Thus, the length of the transverse common tangent is **12 cm**.