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$$ABCD$$ is a quadrilateral in the first quadrant where $$A=(3,9)$$, $$B=(1,1)$$, $$C=(5,3)$$ and $$D=(p,q)$$. The quadrilateral formed by joining the midpoints of $$AB,BC,CD$$ and $$DA$$ is a square. Then $$p+q$$ is
Correct Answer: 10
By Varignon's theorem, the midpoint quadrilateral has sides parallel to the diagonals of $$ABCD$$. Since it is a square, the diagonals $$AC$$ and $$BD$$ are perpendicular and equal. The conditions give $$3q-p=2$$ and $$40=(p-1)^2+(q-1)^2$$, whose first-quadrant solution is $$p=7,q=3$$. Hence $$p+q=10$$.
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