check the picture below.
so if each side of square A is x long, its area is then x*x or namely x².
now, we know the diagonal of square B is x long, so let's find how long is each side, namely how long is y.
[tex] \bf \stackrel{\textit{using the pythagorean theorem}}{c^2=a^2+b^2\implies x^2=y^2+y^2}\implies x=\sqrt{y^2+y^2}\implies x=\sqrt{2y^2}
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x=y\sqrt{2}\implies \boxed{\cfrac{x}{\sqrt{2}}=y}
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\textit{therefore the area of square B is }\cfrac{x}{\sqrt{2}}\cdot \cfrac{x}{\sqrt{2}}\implies \cfrac{x^2}{(\sqrt{2})^2}\implies \cfrac{x^2}{2}\\\\
------------------------------- [/tex]
[tex] \bf \cfrac{\textit{area of B}}{\textit{area of A}}\implies \cfrac{\quad \frac{x^2}{2}\quad }{x^2}\implies \cfrac{\quad \frac{x^2}{2}\quad }{\frac{x^2}{1}}\implies \cfrac{x^2}{2}\cdot \cfrac{1}{x^2}\implies \cfrac{1}{2} [/tex]
