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. Nothing definite can be said about the relation among `R_(1), R_(2)` and R.

Explanation

To solve the problem, we need to establish the relationship between the circumferences of the circles with given radii. Here’s a step-by-step breakdown of the solution:

### Step 1: Write the formula for the circumference of a circle
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

### Step 2: Write the circumferences for the two circles
For the two circles with radii \( R_1 \) and \( R_2 \), the circumferences are:
- Circumference of the first circle:
\[ C_1 = 2\pi R_1 \]
- Circumference of the second circle:
\[ C_2 = 2\pi R_2 \]

### Step 3: Write the equation for the sum of the circumferences
According to the problem, the sum of the circumferences of the two circles is equal to the circumference of a circle with radius \( R \):
\[ C_1 + C_2 = C \]
Substituting the expressions for \( C_1 \), \( C_2 \), and \( C \):
\[ 2\pi R_1 + 2\pi R_2 = 2\pi R \]

### Step 4: Factor out \( 2\pi \)
We can factor \( 2\pi \) out from the left-hand side:
\[ 2\pi (R_1 + R_2) = 2\pi R \]

### Step 5: Cancel \( 2\pi \) from both sides
Since \( 2\pi \) is common on both sides of the equation, we can cancel it:
\[ R_1 + R_2 = R \]

### Conclusion
Thus, we conclude that:
\[ R = R_1 + R_2 \]