If the sum of the circumferences of two circles with radii `R_1 and R_2` is equal to the circumference of a circle of radius R, then

Options

A. `R_1 + R_2 = R`
B. `R_1 + R_2 gt R`
C. `R_1 + R_2 lt R`
D. `R_1^2 + R_2^2lt R`

Answer

To solve the problem, we need to use the formula for the circumference of a circle, which is given by:

\[ C = 2\pi R \]

where \( R \) is the radius of the circle.

### Step-by-Step Solution:

1. **Calculate the Circumference of the Two Circles:**
- The circumference of the first circle with radius \( R_1 \) is:
\[ C_1 = 2\pi R_1 \]
- The circumference of the second circle with radius \( R_2 \) is:
\[ C_2 = 2\pi R_2 \]

2. **Sum the Circumferences:**
- The total circumference of the two circles is:
\[ C_{total} = C_1 + C_2 = 2\pi R_1 + 2\pi R_2 = 2\pi (R_1 + R_2) \]

3. **Circumference of the Circle with Radius \( R \):**
- The circumference of the circle with radius \( R \) is:
\[ C = 2\pi R \]

4. **Set the Two Circumferences Equal:**
- According to the problem, the sum of the circumferences of the two circles is equal to the circumference of the circle with radius \( R \):
\[ 2\pi (R_1 + R_2) = 2\pi R \]

5. **Simplify the Equation:**
- We can divide both sides by \( 2\pi \) (assuming \( \pi \neq 0 \)):
\[ R_1 + R_2 = R \]

### Conclusion:
From the above steps, we conclude that the sum of the radii \( R_1 + R_2 \) is equal to the radius \( R \).