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
\(R_1 + R_2 = R\)
\(R_1^2 + R_2^2 = R^2\)
\(R_1 + R_2 < R\)
\(R_1^2 + R_2^2 < R^2\)
Step 1: Recall the formula for the area of a circle.
Area of a circle = \(\pi r^2\), where \(r\) is the radius.
Step 2: Find the area of the first circle.
Radius = \(R_1\). So, Area = \(\pi R_1^2\).
Step 3: Find the area of the second circle.
Radius = \(R_2\). So, Area = \(\pi R_2^2\).
Step 4: Add the two areas.
Total area = \(\pi R_1^2 + \pi R_2^2\).
Step 5: According to the question, this total area is equal to the area of a bigger circle with radius \(R\).
Area of bigger circle = \(\pi R^2\).
Step 6: Write the equation.
\(\pi R_1^2 + \pi R_2^2 = \pi R^2\).
Step 7: Cancel \(\pi\) from both sides (since it is common and non-zero).
\(R_1^2 + R_2^2 = R^2\).
Final Answer: The correct option is (B).