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Question Number 72603 by aliesam last updated on 30/Oct/19

Answered by mr W last updated on 30/Oct/19

Commented by aliesam last updated on 30/Oct/19

$$thankyousir$$

Commented by mr W last updated on 30/Oct/19

$$sidelengthofsquarel$$$$l=2a\cos \theta$$$$l=\frac{\sqrt{3}a}{2}+\sqrt{2}a\cos (\frac{\pi }{4}-\theta )=\frac{\sqrt{3}a}{2}+a(\cos \theta +\sin \theta )$$$$\frac{\sqrt{3}a}{2}+a(\cos \theta +\sin \theta )=2a\cos \theta$$$$\frac{\sqrt{3}}{2}=\cos \theta -\sin \theta =\sqrt{2}\sin (\frac{\pi }{4}-\theta )$$$$\frac{\sqrt{6}}{4}=\sin (\frac{\pi }{4}-\theta )$$$$\Rightarrow \theta =\frac{\pi }{4}-{\sin }^{-1}\frac{\sqrt{6}}{4}$$$$Y=\frac{\sqrt{3}}{2}a-a\sin \theta =\frac{\sqrt{3}}{2}a-a\left(\frac{\sqrt{5}-\sqrt{3}}{4}\right)=\frac{3\sqrt{3}-\sqrt{5}}{4}a$$$$X=2a\cos \theta -Y=2a\left(\frac{\sqrt{5}+\sqrt{3}}{4}\right)-\frac{3\sqrt{3}-\sqrt{5}}{4}a$$$$X=\frac{3\sqrt{5}-\sqrt{3}}{4}a$$$$\frac{X}{Y}=\frac{3\sqrt{5}-\sqrt{3}}{3\sqrt{3}-\sqrt{5}}=\frac{4\sqrt{15}+3}{11}\approx 1.681$$

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