Two circles with radii 25 cm and 9 cm touch each other externally. Find the length of the direct common tangent.

Options

A. 32 cm
B. 50 cm
C. 40 cm
D. 30 cm

Answer

To find the length of the direct common tangent between two circles with radii 25 cm and 9 cm that touch each other externally, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Circles:**
- Let \( r_1 = 25 \) cm (radius of the larger circle).
- Let \( r_2 = 9 \) cm (radius of the smaller circle).

2. **Use the Formula for the Length of the Direct Common Tangent:**
- The length \( L \) of the direct common tangent between two circles can be calculated using the formula:
\[ L = \sqrt{r_1 \cdot r_2 \cdot 2} \]

3. **Substitute the Values into the Formula:**
- Substitute \( r_1 \) and \( r_2 \) into the formula:
\[ L = \sqrt{25 \cdot 9 \cdot 2} \]

4. **Calculate the Product Inside the Square Root:**
- First, calculate \( 25 \cdot 9 = 225 \).
- Then, multiply by 2:
\[ 225 \cdot 2 = 450 \]

5. **Take the Square Root:**
- Now, calculate the square root of 450:
\[ L = \sqrt{450} \]
- We can simplify \( \sqrt{450} \):
\[ \sqrt{450} = \sqrt{225 \cdot 2} = \sqrt{225} \cdot \sqrt{2} = 15\sqrt{2} \]

6. **Approximate the Value:**
- To find the approximate length, we can use \( \sqrt{2} \approx 1.414 \):
\[ L \approx 15 \cdot 1.414 \approx 21.21 \text{ cm} \]

### Final Answer:
The length of the direct common tangent is approximately \( 21.21 \) cm.