Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.2 AIME, harder Prove it United States

Problem:
A parallelogram $P$ can be folded over a straight line so that the resulting shape is a regular pentagon with side length $1$ . Compute the perimeter of $P$ .

Solution

Solution:

In regular pentagon $ABCDE$ (labeled clockwise), reflect $ABDE$ across $AB$ to obtain $ABD'E'$ . Then, $CDE'D'$ is one such parallelogram $P$ . The length of $CD'$ is
$CB + BD = 1 + 2\cos \angle CBD = 1 + 2\cos (\pi /5) = 1 + \frac{\sqrt{5} + 1}{2} = \frac{\sqrt{5} + 3}{2}.$
Hence, the perimeter of the desired parallelogram is
$2\left(1 + \frac{\sqrt{5} + 3}{2}\right) = 5 + \sqrt{5}.$

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