The distance between the centres of two circles with radii 9 cm and 16 cm is 25 cm. The length of the segment of the tan gent between them is

Options

A. 24 cm
B. 25 cm
C. `50/3 cm`
D. 12 cm

Answer

To find the length of the segment of the tangent between two circles, we can use the formula for the length of the direct common tangent between two circles. The formula is given by:

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

where:
- \(L\) is the length of the tangent,
- \(d\) is the distance between the centers of the two circles,
- \(r_1\) is the radius of the first circle,
- \(r_2\) is the radius of the second circle.

### Step-by-Step Solution:

1. **Identify the given values:**
- Radius of the first circle, \(r_1 = 9 \, \text{cm}\)
- Radius of the second circle, \(r_2 = 16 \, \text{cm}\)
- Distance between the centers of the circles, \(d = 25 \, \text{cm}\)

2. **Calculate the difference of the radii:**
\[ r_1 - r_2 = 9 - 16 = -7 \, \text{cm} \]

3. **Square the difference of the radii:**
\[ (r_1 - r_2)^2 = (-7)^2 = 49 \, \text{cm}^2 \]

4. **Square the distance between the centers:**
\[ d^2 = 25^2 = 625 \, \text{cm}^2 \]

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

6. **Calculate the value inside the square root:**
\[ 625 - 49 = 576 \]

7. **Take the square root to find the length of the tangent:**
\[ L = \sqrt{576} = 24 \, \text{cm} \]

### Final Answer:
The length of the segment of the tangent between the two circles is **24 cm**.