jmc

Since $A=(u,v)$, we can find the coordinates of the other points: $B=(v,u)$, $C=(-v,u)$, $D=(-v,-u)$, $E=(v,-u)$. If we graph those points, we notice that since the latter four points are all reflected across the x/y-axis, they form a rectangle, and $ABE$ is a triangle. The area of $BCDE$ is $(2u)(2v)=4uv$ and the area of $ABE$ is $\frac{1}{2}(2u)(u-v)=u^2-uv$. Adding these together, we get $u^2+3uv=u(u+3v)=451=11\cdot 41$. Since $u,v$ are positive, $u+3v>u$, and by matching factors we get either $(u,v)=(1,150)$ or $(11,10)$. Since $v<u$ the latter case is the answer, and $u+v=\boxed{21}$.