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Geometry Difficulty 4.9 AIME Prove it United States

Problem:

Triangle $A B C$ has a right angle at $C$ , and $D$ is the foot of the altitude from $C$ to $A B$ . Points $L$ , $M$ , and $N$ are the midpoints of segments $A D$ , $D C$ , and $C A$ , respectively. If $C L = 7$ and $B M = 12$ , compute $B N^{2}$ .

Solution

Solution:

Note that $C L$ , $B M$ , and $B N$ are corresponding segments in the similar triangles $\triangle A C D \sim \triangle C B D \sim \triangle A B C$ . So, we have
$C L : B M : B N = A D : C D : A C$
Since $A D^{2} + C D^{2} = A C^{2}$ , we also have $C L^{2} + B M^{2} = B N^{2}$ , giving an answer of $49 + 144 = 193$ .

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