Hint: Pick any apex of the hexagon, point A. Let B be one of the 2 apices such that AB is a short diagonal. Let C be the apex such that AC is a long diagonal. BC will be a side of the hexagon.
Triangle ABC will be a right angle triangle. Long diagonal AC will be the hypotenuse. Use Pythagorus' rule to relate the lengths.
The interior angle of a regular hexagon is 120°. Use the law of cosines and regularity of the hexagon to express the length of the short diagonal to the side of the hexagon.
Solve these simultaneous equations.
In a regular hexagon, what is the ratio of the length of the shortest diagonal to the length of the longest diagonal? Express your answer as a common fraction in the simplest radical form.
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