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Problem 1029

AMC 12 late, AIME early

Geometry Difficulty 4.9 Prove it Bay Area Mathematical Olympiad · United States

Let triangle $A B C$ have a right angle at $C$ , and let $M$ be the midpoint of the hypotenuse $A B$ . Choose a point $D$ on line $B C$ so that angle $C D M$ measures $30$ degrees. Prove that the segments $A C$ and $M D$ have equal lengths.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

Drop the perpendicular from $M$ to $B C$ . Let $P$ be the point where this perpendicular meets the line $B C$ . Since $\angle M P B$ and $\angle A C B$ are both right angles, and triangle $M P B$ and triangle $A C B$ share the angle at $B$ , triangle $M P B$ and triangle $A C B$ are similar. Since the length of $M B$ is half the length of $B A$ , side $M P$ must be half the length of $A C$ . But triangle $M P D$ is a $30$ - $60$ - $90$ triangle, so the length of $M P$ is also half the length of $M D$ . Therefore, the length of $A C$ equals the length of $M D$ .

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