Express `R_(3)`interms of `R_(1) and R_(2),` where the sum of areas of two circles with radii `R _(1) and R_(2)` is equal to the area of the circle of radius `R_(3).`

Options

A. `R_(3)^(2) + R _(2) ^(2) = R _(1) ^(2)`
B. `R _(3) ^(2) = R _(1) ^(2) - R _(2) ^(2)`
C. `R _(3) ^(2) = R _(1) ^(2) + R _(2) ^(2)`
D. `R _(3) ^(2) + R _(1) ^(2) = R _(2) ^(2)`

Answer

To express \( R_3 \) in terms of \( R_1 \) and \( R_2 \), we start by using the formula for the area of a circle, which is given by:

\[ \text{Area} = \pi r^2 \]

1. **Calculate the area of the first circle with radius \( R_1 \)**:
\[ \text{Area}_1 = \pi R_1^2 \]

2. **Calculate the area of the second circle with radius \( R_2 \)**:
\[ \text{Area}_2 = \pi R_2^2 \]

3. **Set up the equation based on the problem statement**:
According to the problem, the sum of the areas of the two circles is equal to the area of the circle with radius \( R_3 \):
\[ \text{Area}_1 + \text{Area}_2 = \text{Area}_3 \]
This can be written as:
\[ \pi R_1^2 + \pi R_2^2 = \pi R_3^2 \]

4. **Factor out \( \pi \)** from the left side of the equation**:
\[ \pi (R_1^2 + R_2^2) = \pi R_3^2 \]

5. **Cancel \( \pi \) from both sides** (assuming \( \pi \neq 0 \)):
\[ R_1^2 + R_2^2 = R_3^2 \]

6. **Take the square root of both sides to solve for \( R_3 \)**:
\[ R_3 = \sqrt{R_1^2 + R_2^2} \]

Thus, we have expressed \( R_3 \) in terms of \( R_1 \) and \( R_2 \):
\[ R_3 = \sqrt{R_1^2 + R_2^2} \]