Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Prove it Italy

Problem:

In a cube of side $12$ , $P$ and $Q$ are the centers of two faces that share the edge $AB$ . What is the volume of the tetrahedron having as vertices the points $A$ , $B$ , $P$ , $Q$ ?

Solution

Solution:

The answer is $72$ . Indeed the tetrahedron $ABPQ$ can be thought of as a pyramid with base $ABP$ and height $QM$ , where $M$ is the midpoint of $AB$ . If the side of the cube is $12$ , the base area will be $\frac{144}{4} = 36$ and the height will be $\frac{12}{2} = 6$ . The volume of the tetrahedron will thus be $\frac{1}{3} \cdot 36 \cdot 6 = 72$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.