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Question Number 68961 by ahmadshah last updated on 17/Sep/19

Answered by mr W last updated on 17/Sep/19

Commented by mr W last updated on 17/Sep/19

$$sidelengthofsquare=4+x$$$$A_{green}=A_{orange}=\frac{x(4+x)}{2}$$$$A_{white}=A_{square}-3A_{green}$$$$={(4+x)}^2-\frac{3x(4+x)}{2}$$$$=\frac{(4+x)(8-x)}{2}=\frac{(4+x)BC}{2}$$$$\Rightarrow BC=8-x$$$$AB=(4+x)-BC=2x-4$$$$A_{orange}=\frac{(4+x)BD}{2}=\frac{x(4+x)}{2}$$$$\Rightarrow BD=x=AE$$$$\frac{AB}{AE}=\frac{x}{4+x}$$$$\Rightarrow \frac{2x-4}{x}=\frac{x}{4+x}$$$$(2x-4)(4+x)=x^2$$$$x^2+4x-16=0$$$$\Rightarrow x=2(\sqrt{5}-1)$$$$L_{blue}=\sqrt{x^2+{(4+x)}^2}=\sqrt{2(x^2+4x+8)}$$$$=\sqrt{2(16+8)}=4\sqrt{3}$$

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