Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 12B Problems/Problem 10"

(→‎Solution)
(→‎Solution)
Line 11: Line 11:
 
==Solution==
 
==Solution==
  
−
Consider triangle <math>BCG</math>. <math>BG</math> = 1 and <math>BC</math> = 2. <math>\angle CBG = 120 ^{\circ}</math> because it is an interior angle of a regular hexagon. By the [[Law of Cosines]], we have:  
+
Consider triangle <math>BCG</math>. <math>BG = 1</math> and <math>BC = 2</math>. <math>\angle CBG = 120 ^{\circ}</math> because it is an interior angle of a regular hexagon. By the [[Law of Cosines]], we have:  
  
−
<cmath> CG^2 = BG^2 + BC^2 - 2 \cdot BG \cdot BC \cdot \cos \angle CBG \\ CG^2 = 1^2 + 2^2 - 2 \cdot 1 \cdot 2 \cdot \cos 120 ^{\circ} \\ CG = \sqrt 7.</cmath>
+
<cmath> CG^2 = BG^2 + BC^2 - 2 \cdot BG \cdot BC \cdot \cos \angle CBG</cmath>
 +
 
 +
<cmath>CG^2 = 1^2 + 2^2 - 2 \cdot 1 \cdot 2 \cdot \cos 120 ^{\circ}</cmath>
 +
 
 +
<cmath>CG = \sqrt 7.</cmath>
  
 
By [[SAS Congruence]], triangles <math>BCG</math>, <math>CDH</math>, <math>HEF</math>, and <math>AFG</math> are congruent, and by [[CPCTC]], quadrilateral <math>GCHF</math> is a rhombus. Therefore, its perimeter is <math>4 \cdot CG = \boxed{\textbf{(D)} \ 4 \sqrt 7}</math>.
 
By [[SAS Congruence]], triangles <math>BCG</math>, <math>CDH</math>, <math>HEF</math>, and <math>AFG</math> are congruent, and by [[CPCTC]], quadrilateral <math>GCHF</math> is a rhombus. Therefore, its perimeter is <math>4 \cdot CG = \boxed{\textbf{(D)} \ 4 \sqrt 7}</math>.

Revision as of 18:45, 17 November 2022

Problem

Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$?

$\textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4\sqrt5 \qquad \textbf{(D)}\ 4\sqrt7 \qquad \textbf{(E)}\ 12$

Solution

Consider triangle $BCG$. $BG = 1$ and $BC = 2$. $\angle CBG = 120 ^{\circ}$ because it is an interior angle of a regular hexagon. By the Law of Cosines, we have:

\[CG^2 = BG^2 + BC^2 - 2 \cdot BG \cdot BC \cdot \cos \angle CBG\]

\[CG^2 = 1^2 + 2^2 - 2 \cdot 1 \cdot 2 \cdot \cos 120 ^{\circ}\]

\[CG = \sqrt 7.\]

By SAS Congruence, triangles $BCG$, $CDH$, $HEF$, and $AFG$ are congruent, and by CPCTC, quadrilateral $GCHF$ is a rhombus. Therefore, its perimeter is $4 \cdot CG = \boxed{\textbf{(D)} \ 4 \sqrt 7}$.

See Also

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_12B_Problems/Problem_10&oldid=182012"