If the distance between the centres of two circles of radio 3,4 is 25 then the length of the tranverse common tangent is

Options

A. 24
B. 12
C. 26
D. 13

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**:
- Radius of the first circle, \(r_1 = 3\)
- Radius of the second circle, \(r_2 = 4\)
- Distance between the centers of the circles, \(d = 25\)

2. **Calculate the sum of the radii**:
\[ r_1 + r_2 = 3 + 4 = 7 \]

3. **Square the distance \(d\)**:
\[ d^2 = 25^2 = 625 \]

4. **Square the sum of the radii**:
\[ (r_1 + r_2)^2 = 7^2 = 49 \]

5. **Substitute these values into the formula**:
\[ L = \sqrt{d^2 - (r_1 + r_2)^2} = \sqrt{625 - 49} \]

6. **Calculate the difference**:
\[ 625 - 49 = 576 \]

7. **Take the square root**:
\[ L = \sqrt{576} = 24 \]

### Final Answer:
The length of the transverse common tangent is \(24\) cm.