The radii of two circles are 12 cm and 5 cm respectively . The radius of the circle, which has circumference equal to the sum of the circumferences of the two circles, is `:`

Options

A. 189 cm
B. 7 cm
C. 17 cm
D. 119 cm

Answer

To solve the problem, we need to find the radius of a new circle whose circumference is equal to the sum of the circumferences of two given circles with radii 12 cm and 5 cm.

### Step-by-Step Solution:

1. **Calculate the Circumference of the First Circle:**
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
For the first circle with radius \( r_1 = 12 \) cm:
\[ C_1 = 2\pi \times 12 = 24\pi \text{ cm} \]

2. **Calculate the Circumference of the Second Circle:**
For the second circle with radius \( r_2 = 5 \) cm:
\[ C_2 = 2\pi \times 5 = 10\pi \text{ cm} \]

3. **Find the Sum of the Circumferences:**
Now, we need to find the total circumference of the two circles:
\[ C_{\text{total}} = C_1 + C_2 = 24\pi + 10\pi = 34\pi \text{ cm} \]

4. **Set the Circumference of the New Circle Equal to the Total Circumference:**
Let the radius of the new circle be \( r \). The circumference of the new circle is:
\[ C_{\text{new}} = 2\pi r \]
We set this equal to the total circumference:
\[ 2\pi r = 34\pi \]

5. **Solve for the Radius \( r \):**
To find \( r \), we can divide both sides of the equation by \( 2\pi \):
\[ r = \frac{34\pi}{2\pi} = \frac{34}{2} = 17 \text{ cm} \]

### Final Answer:
The radius of the circle, which has a circumference equal to the sum of the circumferences of the two circles, is **17 cm**.