2001 AMC 12 Problem 20

The midpoints are $M = (2, 5)$ of $\overline{AB}$ and $N = (3, 2)$ of $\overline{BC}.$ For the midpoint quadrilateral to be a square, consecutive sides are perpendicular and equal. With $\overrightarrow{NM} = \langle -1, 3 \rangle,$ the side $\overrightarrow{MQ}$ to the midpoint $Q$ of $\overline{DA}$ must be $\langle 3, 1 \rangle,$ so $Q = (5, 6).$ Since $Q$ is the midpoint of $\overline{DA}$ and $A = (3, 9),$ we get $D = 2Q - A = (7, 3).$ The sum of its coordinates is $7 + 3 = 10.$ Thus, the correct answer is C.