Question

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

• A. $R_1+R_2=R$
• B. $R_1^2+R_2^2=R^2$
• C. $R_1+R_2<R$
• D. $R_1^2+R_2^2<R^2$

Answer 1

The correct option is B

$R_1^2+R_2^2=R^2$

Explanation for correct option:

Area of circle:

For a circle of radius $R$ the area $A$ is given by $A=\pi R^2$

Therefore for a circle of radius $R_1$ the area $A_1$ is given by $A_1=\pi R_1^2$ and the circle of radius $R_2$ the area $A_2$ is $A_2=\pi R_2^2$

Given that 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 $R$

$$\begin{gathered} \therefore A_1+A_2=A \\ \Rightarrow \pi R_1^2+\pi R_2^2=\pi R_{}^2 \\ \Rightarrow R_1^2+R_2^2=R_{}^2 \end{gathered}$$

Hence, option (B) is correct.