2021 Fall AMC 12A Problems/Problem 14
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Solution (Law of Cosines and Equilateral Triangle Area)
Isosceles triangles ,
, and
are identical by SAS similarity.
by CPCTC, and 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, the square of the side length of equilateral triangle BDF is . Hence, the area of the equilateral triangle is
.
So, the total area of the hexagon is the area of the equilateral triangle plus thrice the area of each isosceles triangle or . The perimeter is
.