Vector Dot Products

(2015 Bay Area Mathematical Olympiad-8, Problem D) In a quadrilateral, the two segments connecting the midpoints of its opposite sides are equal in length. Prove that the diagonals of the quadrilateral are perpendicular. (In other words, let $M, N, P,$ and $Q$ be the midpoints of sides $AB, BC, CD,$ and $DA$ in quadrilateral $ABCD.$ It is known that segments $MP$ and $NQ$ are equal in length. Prove that $AC$ and $BD$ are perpendicular.)