Asked by james
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.
Answers
Answered by
Graham
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.
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.
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