2021 Fall AMC 12A Problems/Problem 14
Contents
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
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 will be referring to the following diagram:
Observe that
Letting
the perimeter will be
We know that and using such, we have
Thus, we have
Computing the area of
we have
Plugging back into
we have
which means
and
~ASAB
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.