Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Find the answer

A triangle in the $x y$ -plane is such that when projected onto the $x$ -axis, $y$ -axis, and the line $y=x$ , the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$ , and $(5,5)$ and $(7.5,7.5)$ , respectively. What is the triangle's area?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Sketch the lines $x=1, x=5, y=8, y=13, y=10-x$ , and $y=15-x$ . The triangle has to be contained in the hexagonal region contained in all these lines. If all the projections are correct, every other vertex of the hexagon must be a vertex of the triangle, which gives us two possibilities for the triangle. One of these triangles has vertices at $(2,8),(1,13)$ , and $(5,10)$ , and has an area of $\frac{17}{2}$ . It is easy to check that the other triangle has the same area, so the answer is unique.

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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.