Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Prove it Philippines

Problem:

An equilateral triangle is divided into three congruent trapezoids, as shown in the figure. If the perimeter of each trapezoid is 10+5310+5 \sqrt{3}, what is the side length of the triangle?

Figure 1

Solution

Solution:

Each of the trapezoids will have the following interior angles.

Figure 2

We therefore have an isosceles trapezoid. By the manner in which the trapezoids are tiled, the shorter base and the two legs will all have the same length; let it be aa. The longer base is then equal to a+a2+a2=2aa+\frac{a}{2}+\frac{a}{2}=2a. Therefore
a+a+a+2a=10+535a=10+53a=2+3 \begin{aligned} a+a+a+2a & = 10+5\sqrt{3} \\ 5a & = 10+5\sqrt{3} \\ a & = 2+\sqrt{3} \end{aligned}
The side length of a triangle is a+2a=3a=6+33a+2a=3a=6+3\sqrt{3}.

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