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Question Number 81421 by ajfour last updated on 12/Feb/20

Commented by ajfour last updated on 12/Feb/20

If both rectangles have unit  areas, find the side length of  the outer square.

$${If}\:{both}\:{rectangles}\:{have}\:{unit} \\ $$$${areas},\:{find}\:{the}\:{side}\:{length}\:{of} \\ $$$${the}\:{outer}\:{square}. \\ $$

Answered by ajfour last updated on 12/Feb/20

(s−x)x=1  (x(√2)){s(√2)−2(√2)x}=1  equating  2s−4x=s−x  ⇒  x=s/3  ⇒   (((2s)/3))((s/3))=1     ⇒   s=((3(√2))/2) .

$$\left({s}−{x}\right){x}=\mathrm{1} \\ $$$$\left({x}\sqrt{\mathrm{2}}\right)\left\{{s}\sqrt{\mathrm{2}}−\mathrm{2}\sqrt{\mathrm{2}}{x}\right\}=\mathrm{1} \\ $$$${equating} \\ $$$$\mathrm{2}{s}−\mathrm{4}{x}={s}−{x} \\ $$$$\Rightarrow\:\:{x}={s}/\mathrm{3} \\ $$$$\Rightarrow\:\:\:\left(\frac{\mathrm{2}{s}}{\mathrm{3}}\right)\left(\frac{{s}}{\mathrm{3}}\right)=\mathrm{1}\:\:\: \\ $$$$\Rightarrow\:\:\:{s}=\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{2}}\:. \\ $$

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