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Geometry Difficulty 3.6 AMC 10/12 Find the answer Slovenia

Take an equilateral triangle $ABC$ with the side of length $a$ . Draw the squares $BADE$ and $CBFG$ on the sides $AB$ and $BC$ , respectively (see figure). How long is the line segment $DG$ ?

Pick one

Solution

Denote the intersections of the segment $DG$ with the lines $AB$ and $BC$ by $K$ and $L$ respectively. Due to symmetry the segment $DG$ is parallel to the segment $AC$ , so $KBL$ is an equilateral triangle. The triangles $DKA$ and $LGC$ are one half of an equilateral triangle with altitudes of length $a$ and sides $|DK| = |GL| = \frac{2a}{\sqrt{3}}$ . The side of the equilateral triangle $KBL$ therefore measures
$|KL| = a - |AK| = a - \frac{|DK|}{2} = a - \frac{a}{\sqrt{3}}.$
The length of the segment $DG$ is

|DG| = 2|DK| + |KL| = $\frac{4a}{\sqrt{3}} + \left(a - \frac{a}{\sqrt{3}}\right)$ = a + $\frac{3a}{\sqrt{3}}$ = (1 + $\sqrt{3})a.$

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