Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$?
$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6$
There are $6$ pairs of vertices of $R$. However, both diagonals determine the same circle, therefore the answer is $\boxed{5}$.

[asy] size(200); defaultpen(0.8);  pair A=(0,0), B=(5,0), C=(5,2), D=(0,2);  draw ( A--B--C--D--cycle ); draw( circle( (A+B)/2, length((A-B)/2) ), red ); draw( circle( (C+B)/2, length((C-B)/2) ), red ); draw( circle( (C+D)/2, length((C-D)/2) ), red ); draw( circle( (A+D)/2, length((A-D)/2) ), red ); draw( circle( (A+B+C+D)/4, length((A-C)/2) ), green ); [/asy]