Question : As shown in the given figure, inside the large semicircle, three semicircles (with equal radii) are drawn so that their diameters all sit on the large semicircle's diameter. What is the ratio between the red and blue areas?
Option 1: 4 : 3
Option 2: 1 : 2
Option 3: 2 : 1
Option 4: 3 : 4
Correct Answer: 1 : 2
Solution :
Let the radius of the smaller semicircle be $r$.
According to the question,
The radius of the bigger semicircle with respect to the smaller circle will be $3r$.
Area of the small semicircle = $\frac{\pi r^2}{2}$
Area of the large semicircle = $\frac {\pi (3r)^2}{2}$ = $\frac{9 \pi r^2}{2}$
Red area = 3 × Area of the small semicircle = $\frac{3 \pi r^2}{2}$
Blue area = Area of the large semicircle – Red area = $\frac{9\pi r^2}{2}– \frac{3\pi r^2}{2} = \frac{6\pi r^2}{2}=3 \pi r^2$
$\therefore\frac{\text{Red Area}}{\text{Blue Area}}$ = $\frac{\frac{3 \pi r^2}{2}}{\frac{6 \pi r^2}{2}}$ = $\frac{1}{2}$
Hence, the correct answer is 1 : 2.
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