Maths Olympiad Prep

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Geometry Difficulty 4.9 AIME Prove it United States

Problem:

The rectangle MNPQM N P Q is inside the rectangle ABCDA B C D. The portion of the rectangle ABCDA B C D outside of MNPQM N P Q is colored in green. Using just a straightedge construct a line that divides the green figure in two parts of equal areas.

Solution

Solution:

The line passing through the centers of ABCDA B C D and MNPQM N P Q divides each of the rectangles in two pieces of equal areas. Hence that line will divide the green part into equal areas. The line can be easily constructed using only a straightedge because we can construct the centers of the given rectangles as intersections of their diagonals.

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