Problem:
Regular hexagon has side length . Let be the midpoint of , and let be the midpoint of . What is the perimeter of ?
Answer Choices:
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Solution:
By symmetry the quadrilateral is a rhombus. Let be the center of the hexagon, which is also the center of the rhombus. Then is a right triangle with shorter leg 1 , so and . Furthermore, is a right triangle, and its legs are and . By the Pythagorean Theorem , and the perimeter of the rhombus is .
Let be the side length of the rhombus . By the Law of Cosines applied to ,
Thus , and the requested perimeter is .
The problems on this page are the property of the MAA's American Mathematics Competitions