If the sum of the areas of two circles with radii `R_(1)` and `R_(2)` is equal to the area of a circle of radius R, then

Options

A. `R_(1) + R_(2) = R`
B. `R_(1)^(2)+R_(2)^(2) = R^(2)`
C. `R_(1) + R_(2) lt R`
D. `R_(1)^(2) + R_(2)^(2) lt R^(2)`

Explanation

To solve the problem, we need to establish the relationship between the areas of the circles based on the given information.

### Step-by-Step Solution:

1. **Understand the Area of a Circle**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

2. **Write the Areas of the Given Circles**:
For the two circles with radii \( R_1 \) and \( R_2 \), the areas can be expressed as:
\[ \text{Area of Circle 1} = \pi R_1^2 \]
\[ \text{Area of Circle 2} = \pi R_2^2 \]

3. **Sum the Areas of the Two Circles**:
The sum of the areas of the two circles is:
\[ \text{Total Area} = \pi R_1^2 + \pi R_2^2 \]

4. **Area of the Circle with Radius \( R \)**:
The area of the circle with radius \( R \) is:
\[ \text{Area of Circle with radius } R = \pi R^2 \]

5. **Set Up the Equation**:
According to the problem, the sum of the areas of the two circles is equal to the area of the circle with radius \( R \):
\[ \pi R_1^2 + \pi R_2^2 = \pi R^2 \]

6. **Factor Out \( \pi \)**:
We can factor \( \pi \) from the left-hand side:
\[ \pi (R_1^2 + R_2^2) = \pi R^2 \]

7. **Cancel \( \pi \)**:
Since \( \pi \) is common on both sides, we can cancel it out (assuming \( \pi \neq 0 \)):
\[ R_1^2 + R_2^2 = R^2 \]

8. **Conclusion**:
Thus, we have established that:
\[ R^2 = R_1^2 + R_2^2 \]

### Final Result:
The relationship between the radii is given by:
\[ R^2 = R_1^2 + R_2^2 \]