Question Number 127740 by Algoritm last updated on 01/Jan/21
Commented by Algoritm last updated on 01/Jan/21
$$\mathrm{yellow}\:\mathrm{shaded}\:\mathrm{area}=? \\ $$
Commented by mr W last updated on 02/Jan/21
$${do}\:{you}\:{have}\:{the}\:{correct}\:{answer}? \\ $$$${i}\:{got}\:{shaded}\:{area}=\frac{{area}\:{of}\:{square}}{\mathrm{24}}. \\ $$
Commented by mr W last updated on 02/Jan/21
$${why}\:{don}'{t}\:{you}\:{answer}\:{this}\:{simple} \\ $$$${question}\:{from}\:{me}? \\ $$
Commented by Algoritm last updated on 02/Jan/21
$$\mathrm{yes}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 02/Jan/21
$${thank}\:{you}\:{sir}! \\ $$
Answered by mr W last updated on 01/Jan/21
Commented by mr W last updated on 01/Jan/21
$$\mathrm{tan}\:\alpha=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{tan}\:\mathrm{2}\alpha=\frac{\mathrm{2}×\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{1}−\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }=\frac{\mathrm{4}}{\mathrm{3}} \\ $$$$\beta=\pi−\mathrm{2}×\mathrm{2}\alpha=\pi−\mathrm{4}\alpha \\ $$$$\mathrm{sin}\:\beta=\mathrm{sin}\:\mathrm{4}\alpha=\frac{\mathrm{2}×\frac{\mathrm{4}}{\mathrm{3}}}{\mathrm{1}+\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{2}} }=\frac{\mathrm{24}}{\mathrm{25}} \\ $$$${shaded}\:{area}=\left(\frac{{a}}{\mathrm{4}}×\mathrm{sin}\:\mathrm{2}\alpha\right)^{\mathrm{2}} ×\frac{\mathrm{1}}{\mathrm{sin}\:\beta} \\ $$$$=\frac{{a}^{\mathrm{2}} }{\mathrm{16}}×\left(\frac{\mathrm{4}}{\mathrm{5}}\right)^{\mathrm{2}} ×\frac{\mathrm{25}}{\mathrm{24}}=\frac{{a}^{\mathrm{2}} }{\mathrm{24}} \\ $$