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Slovenia — National Math Olympiad

Geometry Difficulty 4.3 AIME Find the answer Slovenia

A square with the sides measuring 11 cm is divided into three triangles. Which of the following statements is certainly true?

Pick one

Solution

There are only three essentially different ways of dividing a square into three triangles:

Figure 1

Figure 2

Figure 3

both cuts run from a vertex to a common point on one of the sides.
Notice that in all three cases the area of the biggest of the three triangles is 0.5 cm20.5 \text{ cm}^2. The counterexamples for the statements A, C, D and E are ▱A, ▱B and ▱C respectively.

Figure 4

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