If the sum of the areas of two circles with radii R1 and R2 is equal to the area of a circle of radius R, then
Question:
If the sum of the areas of two circles with radii R1 and R2 is equal to the area of a circle of radius R, then
(a) $R_{1}+R_{2}=R$
(b) $R_{1}+R_{2}
(c) $R_{1}^{2}+R_{2}^{2}
(d) $R_{1}^{2}+R_{2}^{2}=R^{2}$
Solution:
(d) $R_{1}^{2}+R_{2}^{2}=R^{2}$
Because the sum of the areas of two circles with radii $R_{1}$ and $R_{2}$ is equal to the area of a circle with radius $R$, we have:
$\pi R_{1}^{2}+\pi R_{2}^{2}=\pi R^{2}$
$\Rightarrow \pi\left(\mathrm{R}_{1}^{2}+\mathrm{R}_{2}^{2}\right)=\pi \mathrm{R}^{2}$
$\Rightarrow \mathrm{R}_{1}^{2}+\mathrm{R}_{2}^{2}=\mathrm{R}^{2}$
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