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Question Number 193754 by Mingma last updated on 19/Jun/23

Answered by talminator2856792 last updated on 19/Jun/23

$$\mathrm{let}R,r\mathrm{represent}\mathrm{radius}\mathrm{of}\mathrm{big}\mathrm{semicircle}$$$$\mathrm{and}\mathrm{small}\mathrm{semicricle},\mathrm{respectively}.$$$$\mathrm{by}\mathrm{pythagorean}\mathrm{theorem}:$$$$R=\sqrt{{\left(\frac{r\sqrt{3}}{2}\right)}^2+{\left(3r\right)}^2}$$$$R=\sqrt{\frac{3r^2}{4}+9r^2}$$$$R=\sqrt{\frac{39r^2}{4}}$$$$R=\frac{r\sqrt{39}}{2}$$$$\mathrm{fraction}\mathrm{of}\mathrm{big}\mathrm{semicircle}\mathrm{shaded}$$$$=3\times \frac{\frac{\pi r^2}{2}}{\frac{\pi R^2}{2}}$$$$=3\times \frac{r^2}{R^2}$$$$=3\times \frac{r^2}{\frac{39r^2}{4}}$$$$=3\times \frac{1}{\frac{39}{4}}$$$$=\frac{12}{39}$$

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