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
Divide the equilateral hexagon into isosceles triangles , , and and triangle . The three isosceles triangles 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 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 of the hexagon 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
Solution 3
We will be using this diagram:
Video Solution by TheBeautyofMath
Cleverly done with Trig https://youtu.be/AgSE7HPCVR0
Video Solution (Logic and Trigonometry)
~Education, the Study of Everything
~IceMatrix
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=g6Dk6An2ALY&t=208s
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 |
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