HMMT February

Geometry Difficulty 4.6 AIME Find the answer

Let $ABCD$ be a convex quadrilateral with $AC=7$ and $BD=17$ . Let $M, P, N, Q$ be the midpoints of sides $AB, BC, CD, DA$ respectively. Compute $MN^{2}+PQ^{2}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

$MPNQ$ is a parallelogram whose side lengths are 3.5 and 8.5 so the sum of squares of its diagonals is $\frac{7^{2}+17^{2}}{2}=169$ .

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