Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Prove it United States

Problem:

Consider a cube $A B C D E F G H$ , where $A B C D$ and $E F G H$ are faces, and segments $A E$ , $B F$ , $C G$ , $D H$ are edges of the cube. Let $P$ be the center of face $E F G H$ , and let $O$ be the center of the cube. Given that $A G=1$ , determine the area of triangle $A O P$ .

Solution

Solution:

Answer: $\frac{\sqrt{2}}{24}$

From $A G=1$ , we get that $A E=\frac{1}{\sqrt{3}}$ and $A C=\frac{\sqrt{2}}{\sqrt{3}}$ . We note that triangle $A O P$ is located in the plane of rectangle $A C G E$ . Since $O P \parallel C G$ and $O$ is halfway between $A C$ and $E G$ , we get that $[A O P]=\frac{1}{8}[A C G E]$ . Hence, $[A O P]=\frac{1}{8}\left(\frac{1}{\sqrt{3}}\right)\left(\frac{\sqrt{2}}{\sqrt{3}}\right)=\frac{\sqrt{2}}{24}$ .

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