The radii of two circles are 12 cm and 7 cm respectively. The radius of the circle which has circumference equal to the sum of the circumference of the two circles is `:`

Options

A. 5 cm
B. 9.2 cm
C. 19cm
D. 17 cm

Answer

To find the radius of the circle whose circumference is equal to the sum of the circumferences of two given circles, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given radii of the circles**:
- The radius of the first circle \( r_1 = 12 \) cm.
- The radius of the second circle \( r_2 = 7 \) cm.

2. **Calculate the circumference of each circle**:
- The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
- For the first circle:
\[ C_1 = 2\pi r_1 = 2\pi \times 12 = 24\pi \text{ cm} \]
- For the second circle:
\[ C_2 = 2\pi r_2 = 2\pi \times 7 = 14\pi \text{ cm} \]

3. **Sum the circumferences of the two circles**:
- The total circumference \( C_{total} \) is:
\[ C_{total} = C_1 + C_2 = 24\pi + 14\pi = 38\pi \text{ cm} \]

4. **Set the total circumference equal to the circumference of the new circle**:
- Let the radius of the new circle be \( r \). Then, the circumference of the new circle is:
\[ C = 2\pi r \]
- We set this equal to the total circumference:
\[ 2\pi r = 38\pi \]

5. **Solve for \( r \)**:
- Divide both sides by \( 2\pi \):
\[ r = \frac{38\pi}{2\pi} = \frac{38}{2} = 19 \text{ cm} \]

### Final Answer:
The radius of the circle which has a circumference equal to the sum of the circumferences of the two circles is **19 cm**.