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, 2019

Geometry Difficulty 5.4 AIME, harder Prove it United States

Problem:

Tessa has a figure created by adding a semicircle of radius $1$ on each side of an equilateral triangle with side length $2$ , with semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points?

Solution

Solution:

Note that both points must be in different semicircles to reach the maximum distance. Let these points be $M$ and $N$ , and $O_{1}$ and $O_{2}$ be the centers of the two semicircles where they lie respectively. Then
$MN \leq MO_{1} + O_{1}O_{2} + O_{2}N$
Note that the right side will always be equal to $3$ ( $MO_{1} = O_{2}N = 1$ from the radius condition, and $O_{1}O_{2} = 1$ from being a midline of the equilateral triangle), hence $MN$ can be at most $3$ . Finally, if the four points are collinear (when $M$ and $N$ are defined as the intersection of line $O_{1}O_{2}$ with the two semicircles), then equality will hold. Therefore, the greatest possible distance between $M$ and $N$ is $3$ .

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