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Question Number 159755 by puissant last updated on 20/Nov/21

Commented by puissant last updated on 20/Nov/21

$$Showthat\frac{X}{Y}=\frac{1+\sqrt{5}}{2}$$

Answered by mr W last updated on 21/Nov/21

$$sayr=radiusofcircle,a=sidelength$$$$ofsquares$$$$r^2=a^2+{\left(\frac{a}{2}\right)}^2=\frac{5a^2}{4}$$$$\Rightarrow r=\frac{\sqrt{5}a}{2}$$$$X=r-\frac{a}{2}=\frac{(\sqrt{5}-1)a}{2}$$$$Y=a-X=\frac{(3-\sqrt{5})a}{2}$$$$\frac{X}{Y}=\frac{\sqrt{5}-1}{3-\sqrt{5}}=\frac{(\sqrt{5}-1)(3+\sqrt{5})}{4}=\frac{\sqrt{5}+1}{2}$$

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