Yin Yang Circle

Diameter ACE is divided at C in the ratio 2:3 The two semicircles, ABC and CDE, divide the circular region into an upper (shaded) region and a lower region. Find the ratio of the area of the upper region to that of the lower region.

Source: Julia Robinson Mathematics Festival

SOLUTION

Suppose without loss of generality that diameter AC=2 units and diameter CE=3 units.
Area of upper region = area of semicircle CDE + area of semicircle AFE\,- area of semicircle ABC
=\frac{1}{2}\pi\!\left (\frac{3}{2}\right )^2+\frac{1}{2}\pi\!\left (\frac{5}{2}\right )^2-\frac{1}{2}\pi\!\left (\frac{2}{2}\right )^2
=\frac{1}{2}\pi\!\left (\frac{9}{4}+\frac{25}{4}-\frac{4}{4}\right )
=\frac{1}{2}\pi\!\left (\frac{30}{4}\right )
=\frac{15\pi}{4}
Area of lower region = area of semicircle ABC + area of semicircle AGE-  area of semicircle CDE
=\frac{1}{2}\pi\!\left (\frac{2}{2}\right )^2+\frac{1}{2}\pi\!\left (\frac{5}{2}\right )^2-\frac{1}{2}\pi\!\left (\frac{3}{2}\right )^2
=\frac{1}{2}\pi\!\left (\frac{4}{4}+\frac{25}{4}-\frac{9}{4}\right )
=\frac{1}{2}\pi\!\left (\frac{20}{4}\right )
=\frac{10\pi}{4}
Ratio of area of upper region to area of lower region
\frac{15\pi}{4}\div \frac{10\pi}{4}
=\frac{15\pi}{4}\times \frac{4}{10\pi}
=\frac{3}{2}

Answer: \frac{3}{2}.

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About mvtrinh

Retired high school math teacher.
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