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 1 (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
.
Solution 2
We denote by the side length.
Following from SAS, .
Hence, . Hence,
is equilateral.
In , by applying the law of cosines, we have
Therefore,
Because ,
.
Therefore, the perimeter of
is
.
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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