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Harvard-MIT November Tournament

Geometry Difficulty 4.8 AIME Prove it United States

Problem:
Let $AMOL$ be a quadrilateral with $AM = 10$ , $MO = 11$ , and $OL = 12$ . Given that the perpendicular bisectors of sides $AM$ and $OL$ intersect at the midpoint of segment $AO$ , find the length of side $LA$ .

Solution

Solution:
Let $D$ be the midpoint of $AM$ and $E$ be the midpoint of $AO$ . Then, we note that $ADE \sim AMO$ , so $M$ is a right angle. Similarly, $L$ is a right angle. Consequently, we get that
$AO^{2} = OM^{2} + AM^{2} \Rightarrow AL = \sqrt{AO^{2} - OL^{2}} = \sqrt{11^{2} + 10^{2} - 12^{2}} = \sqrt{77}.$

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