Question Number 27771 by ajfour last updated on 14/Jan/18
Commented by ajfour last updated on 14/Jan/18
$${If}\:{an}\:{equilateral}\:{triangle}\:{of} \\ $$$${edge}\:{length}\:{a}\:{is}\:{rotated}\:{about} \\ $$$${about}\:{its}\:{vertex}\:{A}\:{by}\:{an}\:{angle}\:\theta \\ $$$${find}\:{the}\:{area}\:{common}\:{in}\:{its} \\ $$$${new}\:{and}\:{previous}\:{positions}. \\ $$$$\left({the}\:{pink}\:{area}\right). \\ $$
Answered by mrW2 last updated on 14/Jan/18
Commented by mrW2 last updated on 14/Jan/18
$$\mathrm{2}\alpha+\theta=\frac{\pi}{\mathrm{3}} \\ $$$$\Rightarrow\alpha=\frac{\pi}{\mathrm{6}}−\frac{\theta}{\mathrm{2}} \\ $$$$\frac{{BP}}{\mathrm{sin}\:\theta}=\frac{{AB}}{\mathrm{sin}\:\left(\pi−\frac{\pi}{\mathrm{3}}−\theta\right)}=\frac{{a}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}+\theta\right)} \\ $$$$\Rightarrow{BP}=\frac{\mathrm{sin}\:\theta}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}+\theta\right)}×{a} \\ $$$${similarly} \\ $$$$\frac{{CM}}{\mathrm{sin}\:\alpha}=\frac{{a}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}+\alpha\right)} \\ $$$$\Rightarrow{CM}=\frac{\mathrm{sin}\:\alpha}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}+\alpha\right)}×{a}=\frac{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{6}}−\frac{\theta}{\mathrm{2}}\right)}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\theta}{\mathrm{2}}\right)}×{a}=\frac{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{6}}−\frac{\theta}{\mathrm{2}}\right)}{\mathrm{cos}\:\frac{\theta}{\mathrm{2}}}×{a} \\ $$$$ \\ $$$${PM}={a}−\frac{\mathrm{sin}\:\theta}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}+\theta\right)}×{a}−\frac{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{6}}−\frac{\theta}{\mathrm{2}}\right)}{\mathrm{cos}\:\frac{\theta}{\mathrm{2}}}×{a} \\ $$$$={a}\left[\mathrm{1}−\frac{\mathrm{sin}\:\theta}{\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{cos}\:\theta+\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{sin}\:\theta}−\frac{\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{sin}\:\frac{\theta}{\mathrm{2}}}{\mathrm{cos}\:\frac{\theta}{\mathrm{2}}}\right] \\ $$$$={a}\left[\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{2}\:\mathrm{tan}\:\theta}{\:\sqrt{\mathrm{3}}+\mathrm{tan}\:\theta}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{tan}\:\frac{\theta}{\mathrm{2}}\right] \\ $$$$ \\ $$$${A}_{\Delta{APM}} =\frac{\mathrm{1}}{\mathrm{2}}×{PM}×\frac{\sqrt{\mathrm{3}}\:{a}}{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}\:{a}^{\mathrm{2}} }{\mathrm{4}}\left[\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{2}\:\mathrm{tan}\:\theta}{\:\sqrt{\mathrm{3}}+\mathrm{tan}\:\theta}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{tan}\:\frac{\theta}{\mathrm{2}}\right] \\ $$$$ \\ $$$$\Rightarrow{A}_{{APMQ}} =\mathrm{2}{A}_{\Delta{APM}} =\frac{\sqrt{\mathrm{3}}\:{a}^{\mathrm{2}} }{\mathrm{4}}\left[\mathrm{1}−\frac{\mathrm{4}\:\mathrm{tan}\:\theta}{\:\sqrt{\mathrm{3}}+\mathrm{tan}\:\theta}+\sqrt{\mathrm{3}}\:\mathrm{tan}\:\frac{\theta}{\mathrm{2}}\right] \\ $$
Commented by ajfour last updated on 15/Jan/18
$${thank}\:{you}\:{sir}.\:{please}\:{view}\:{my} \\ $$$${way}\:{if}\:{you}\:{haven}'{t}. \\ $$
Commented by mrW2 last updated on 16/Jan/18
$${Your}\:{way}\:{is}\:{very}\:{nice}\:{too}.\:{I}\:{didn}'{t}\:{come} \\ $$$${to}\:{it}. \\ $$
Answered by ajfour last updated on 14/Jan/18
Commented by ajfour last updated on 14/Jan/18
$${eq}.\:{of}\:{BP}\:: \\ $$$${y}=−\sqrt{\mathrm{3}}\left({x}−{a}\right) \\ $$$${eq}.\:{of}\:{AP}\::\:\:\:{y}={x}\mathrm{tan}\:\theta \\ $$$${eq}.\:{of}\:{AM}\::\:\:{y}={x}\mathrm{tan}\:\left(\theta+\alpha\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={x}\mathrm{tan}\:\left(\frac{\pi}{\mathrm{6}}+\frac{\theta}{\mathrm{2}}\right) \\ $$$${y}_{{P}} =−\sqrt{\mathrm{3}}\left({x}_{{P}} −{a}\right)={x}_{{P}} \mathrm{tan}\:\theta \\ $$$$\Rightarrow\:{y}_{{P}} =\frac{{a}\sqrt{\mathrm{3}}\mathrm{tan}\:\theta}{\:\sqrt{\mathrm{3}}+\mathrm{tan}\:\theta} \\ $$$${similarly}\:{y}_{{M}} =\frac{{a}\sqrt{\mathrm{3}}\mathrm{tan}\:\left(\frac{\pi}{\mathrm{6}}+\frac{\theta}{\mathrm{2}}\right)}{\:\sqrt{\mathrm{3}}+\mathrm{tan}\:\left(\frac{\pi}{\mathrm{6}}+\frac{\theta}{\mathrm{2}}\right)} \\ $$$$\:\:\:\:=\frac{{a}\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{3}}\left(\frac{\mathrm{1}−\frac{\mathrm{tan}\:\left(\theta/\mathrm{2}\right)}{\:\sqrt{\mathrm{3}}}}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}+\mathrm{tan}\:\frac{\theta}{\mathrm{2}}}\right)+\mathrm{1}} \\ $$$$=\frac{{a}\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{3}}\left[\frac{\sqrt{\mathrm{3}}−\mathrm{tan}\:\left(\theta/\mathrm{2}\right)}{\mathrm{1}+\sqrt{\mathrm{3}}\mathrm{tan}\:\left(\theta/\mathrm{2}\right)}\right]+\mathrm{1}} \\ $$$$=\frac{{a}\sqrt{\mathrm{3}}\left[\mathrm{1}+\sqrt{\mathrm{3}}\mathrm{tan}\:\left(\theta/\mathrm{2}\right)\right]}{\mathrm{4}} \\ $$$${reqd}.\:{area}=\mathrm{2}×\frac{{a}}{\mathrm{2}}\left({y}_{{M}} −{y}_{{P}} \right) \\ $$$$=\frac{{a}^{\mathrm{2}} \sqrt{\mathrm{3}}}{\mathrm{4}}\left[\mathrm{1}+\sqrt{\mathrm{3}}\mathrm{tan}\:\frac{\theta}{\mathrm{2}}−\frac{\mathrm{4tan}\:\theta}{\:\sqrt{\mathrm{3}}+\mathrm{tan}\:\theta}\right]\:. \\ $$