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, 2017

Geometry Difficulty 4.8 AIME Prove it United States

Problem:
Let $ABCD$ be a convex quadrilateral with $AB = 5$ , $BC = 6$ , $CD = 7$ , and $DA = 8$ . Let $M$ , $P$ , $N$ , $Q$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DA$ respectively. Compute $MN^{2} - PQ^{2}$ .

Solution

Solution:
Draw in the diagonals of the quad and use the median formula three times to get $MN^{2}$ in terms of the diagonals. Do the same for $PQ^{2}$ and subtract, the diagonal length terms disappear and the answer is
$\frac{BC^{2} + DA^{2} - AB^{2} - CD^{2}}{2} = 13$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.