Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

Let $A M O L$ be a quadrilateral with $A M=10, M O=11$ , and $O L=12$ . Given that the perpendicular bisectors of sides $A M$ and $O L$ intersect at the midpoint of segment $A O$ , find the length of side LA.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $D$ be the midpoint of $A M$ and $E$ be the midpoint of $A O$ . Then, we note that $A D E \sim A M O$ , so $M$ is a right angle. Similarly, $L$ is a right angle. Consequently, we get that $A O^{2}=O M^{2}+A M^{2} \Rightarrow A L=\sqrt{A O^{2}-O L^{2}}=\sqrt{11^{2}+10^{2}-12^{2}}=\sqrt{77}$

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