2021 Fall AMC 12A Problems/Problem 14
Problem
In the figure, equilateral hexagon has three nonadjacent acute interior angles that each measure . The enclosed area of the hexagon is . What is the perimeter of the hexagon?
Solution (Law of Cosines and Equilateral Triangle Area)
Isosceles triangles , , and are congruent by SAS congruence. By CPCTC, , so triangle is equilateral.
Let the side length of the hexagon be .
The area of each isosceles triangle is by the fourth formula here.
By the Law of Cosines on triangle , . Hence, the area of the equilateral triangle is .
The total area of the hexagon is thrice the area of each isosceles triangle plus the area of the equilateral triangle, or . Hence, and the perimeter is .
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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