The distance between the centres of two circles is 61 cm and their radii are 35 cm and 24 cm. What is the length (in cm) of the direct common tangent to the circles?

Options

A. 60
B. 54
C. 48
D. 72

Answer

To find the length of the direct common tangent to the two circles, 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 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:**
- Distance between the centers of the circles, \( d = 61 \) cm
- Radius of the first circle, \( r_1 = 35 \) cm
- Radius of the second circle, \( r_2 = 24 \) cm

2. **Calculate the difference of the radii:**
\[ r_1 - r_2 = 35 - 24 = 11 \text{ cm} \]

3. **Substitute the values into the formula:**
\[ L = \sqrt{d^2 - (r_1 - r_2)^2} \]
\[ L = \sqrt{61^2 - 11^2} \]

4. **Calculate \( d^2 \) and \( (r_1 - r_2)^2 \):**
\[ 61^2 = 3721 \]
\[ 11^2 = 121 \]

5. **Subtract the squares:**
\[ L = \sqrt{3721 - 121} \]
\[ L = \sqrt{3600} \]

6. **Calculate the square root:**
\[ L = 60 \text{ cm} \]

### Final answer:
The length of the direct common tangent to the circles is **60 cm**.