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Olimpiadi della Matematica - Gara di Febbraio

Geometry Difficulty 4.4 AIME Find the answer Italy

Problem:

Alice, Berto and Carlo are looking for a treasure. Knowing that the three friends are located at the vertices of an equilateral triangle and that the treasure is located at a point outside the triangle, at a distance of 1 meter from Alice and from Berto and 2 meters from Carlo, how many meters is the side of the triangle?

Pick one

Solution

Solution:

The answer is $(\mathbf{E})$ . Let $A, B, C$ be the three friends and $T$ the treasure. By symmetry $C T$ is perpendicular to $A B$ .

Consider the triangle $A C T$ . $\widehat{A C T}=30^\circ$ and $C T=2 A T$ ; this is enough to conclude that $A C T$ is hemi-equilateral (that is, it is half of an equilateral triangle). In particular $\widehat{C A T}=90^\circ$ and by the Pythagorean theorem $A C=\sqrt{C T^2-A T^2}=\sqrt{2^2-1^2}=\sqrt{3}$ .

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