Difference between revisions of "2021 Fall AMC 12A Problems/Problem 14"
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<math>\textbf{(A)} \: 4 \qquad \textbf{(B)} \: 4\sqrt3 \qquad \textbf{(C)} \: 12 \qquad \textbf{(D)} \: 18 \qquad \textbf{(E)} \: 12\sqrt3</math> | <math>\textbf{(A)} \: 4 \qquad \textbf{(B)} \: 4\sqrt3 \qquad \textbf{(C)} \: 12 \qquad \textbf{(D)} \: 18 \qquad \textbf{(E)} \: 12\sqrt3</math> | ||
− | ==Solution | + | ==Solution (Law of Cosines and Equilateral Triangle Area)== |
Isosceles triangles <math>ABF</math>, <math>CBD</math>, and <math>EDF</math> are congruent by [[Congruent_(geometry)#SAS_Congruence|SAS congruence]]. By [[CPCTC]], <math>BF=BD=DF</math>, so triangle <math>BDF</math> is equilateral. | Isosceles triangles <math>ABF</math>, <math>CBD</math>, and <math>EDF</math> are congruent by [[Congruent_(geometry)#SAS_Congruence|SAS congruence]]. By [[CPCTC]], <math>BF=BD=DF</math>, so triangle <math>BDF</math> is equilateral. |
Revision as of 01:59, 26 November 2021
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
.
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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All AMC 12 Problems and Solutions |
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