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, 2024

Geometry Difficulty 5.7 AIME, harder Prove it United States

Problem:

Inside an equilateral triangle of side length $6$ , three congruent equilateral triangles of side length $x$ with sides parallel to the original equilateral triangle are arranged so that each has a vertex on a side of the larger triangle, and a vertex on another one of the three equilateral triangles, as shown below.

A smaller equilateral triangle formed between the three congruent equilateral triangles has side length $1$ . Compute $x$ .

Solution

Solution:

Let $x$ be the side length of the shaded triangles. Note that the centers of the triangles with side lengths $1$ and $6$ coincide; call this common center $O$ .

The distance from $O$ to a side of the equilateral triangle with side length $1$ is $\frac{\sqrt{3}}{6}$ . Similarly the distance from $O$ to a side of the equilateral triangle with side length $6$ is $\sqrt{3}$ . Notice the difference of these two distances is exactly the length of the altitude of one of shaded triangles. So

$\sqrt{3} - \frac{\sqrt{3}}{6} = \frac{\sqrt{3}}{2} x \Longrightarrow x = \frac{5}{3}$

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