[tex]P_{\Delta1} =\dfrac{1}{2} \cdot a\cdot a=\dfrac{1}{2}a^{2} \\\\P_{\Delta2} =\dfrac{1}{2} \cdot h\cdot a\sqrt{2} =\dfrac{h\cdot a\sqrt{2} }{2}~~\land ~~h=10cm~~\Rightarrow ~~P_{\Delta2} =a5\sqrt{2} \\\\P_{\Delta1} =P_{\Delta2} \\\\\dfrac{1}{2}a^{2}=a5\sqrt{2}\\\\\dfrac{1}{2}a^{2}-a5\sqrt{2}=0\\\\a\cdot ( \dfrac{1}{2}a-5\sqrt{2} )=0\\\\a=0~~\lor ~~\dfrac{1}{2}a-5\sqrt{2}=0\\\\(~~a=0~~\lor~~a=10\sqrt{2}~~)~~\land ~~a>0~~\Rightarrow~~ a=10\sqrt{2}\\\\[/tex]
[tex]Obw_{\Delta} =a+a+a\sqrt{2} =2a+a\sqrt{2} =a\cdot (2+\sqrt{2} )~~\land ~~a=10\sqrt{2}\\\\Obw_{\Delta} =10\sqrt{2}\cdot (2+\sqrt{2} )=20\sqrt{2} +20=20\cdot (\sqrt{2} +1)[/tex]
