Since the perimeter of the rectangle is $40$, we can make the equation $2(x+y)=40$, and then divide both sides by $2$ to get $x+y=20$ Then, since the diagonal is $10 \sqrt{2}$, we can make the equation $x^2+y^2=200$. We can substitute $x=20-y$ into the 2nd equation, and we get $400-40y+y^2+y^2=200$ When we simplify everything we get the equation: $y^2-20y+100=0$ -> $(y-10)^2=0$, so $y=10$ Therefore, $x=10$, so the area of the rectangle is $100$.
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