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

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== Solution 3 ==
 
== Solution 3 ==
  
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[[Image:2022 AMC 12B-10.PNG|thumb|right|509px|Image caption]]
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[[Image:2022 AMC 12B-10.PNG|thumb|none|509px|Image caption]]
 
We use a coordinates approach. Letting the origin be the center of the hexagon, we can let <math>A = (-1, \sqrt{3}), B = (1, \sqrt{3}), C = (2, 0), D = (1, -\sqrt{3}), E = (-1, -\sqrt{3}), F = (-2, 0).</math> Then, <math>G = (0, \sqrt{3})</math> and <math>H = (0, -\sqrt{3}).</math>
 
We use a coordinates approach. Letting the origin be the center of the hexagon, we can let <math>A = (-1, \sqrt{3}), B = (1, \sqrt{3}), C = (2, 0), D = (1, -\sqrt{3}), E = (-1, -\sqrt{3}), F = (-2, 0).</math> Then, <math>G = (0, \sqrt{3})</math> and <math>H = (0, -\sqrt{3}).</math>
  
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<cmath>GC^2 = (0-2)^2 + (\sqrt{3}-0)^2 = 7 \rightarrow GC = \sqrt{7}</cmath>
 
<cmath>GC^2 = (0-2)^2 + (\sqrt{3}-0)^2 = 7 \rightarrow GC = \sqrt{7}</cmath>
  
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Thus, the perimeter of <math>GCHF = CH + HF + FG + GC = \sqrt{7} + \sqrt{7} + \sqrt{7} + \sqrt{7} = \boxed{\textbf{(D)} \ 4\sqrt{7}</math>.
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Thus, the perimeter of <math>GCHF = CH + HF + FG + GC = \sqrt{7} + \sqrt{7} + \sqrt{7} + \sqrt{7} = \boxed{\textbf{(D)} \ 4\sqrt{7}}</math>.
  
 
~sirswagger21
 
~sirswagger21

Revision as of 20:58, 20 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 1 (No trig)

Let the center of the hexagon be $O$. $\triangle AOB$, $\triangle BOC$, $\triangle COD$, $\triangle DOE$, $\triangle EOF$, and $\triangle FOA$ are all equilateral triangles with side length $2$. Thus, $CO = 2$, and $GO = \sqrt{AO^2 - AG^2} = \sqrt{3}$. By symmetry, $\angle COG = 90^{\circ}$. Thus, by the Pythagorean theorem, $CG = \sqrt{2^2 + \sqrt{3}^2} = \sqrt{7}$. Because $CO = OF$ and $GO = OH$, $CG = HC = FH = GF = \sqrt{7}$. Thus, the solution to our problem is $\sqrt{7} + \sqrt{7} + \sqrt{7} + \sqrt{7} = \boxed{\textbf{(D)} \ 4 \sqrt 7}$

~mathboy100

Solution 2

Consider triangle $AFG$. Note that $AF = 2$, $AG = 1$, and $\angle GAF = 120 ^{\circ}$ because it is an interior angle of a regular hexagon. (See note for details.)

By the Law of Cosines, we have:

\begin{align*} FG^2 &= AG^2 + AF^2 - 2 \cdot AG \cdot AF \cdot \cos \angle GAF \\ FG^2 &= 1^2 + 2^2 - 2 \cdot 1 \cdot 2 \cdot \cos 120 ^{\circ} \\ FG^2 &= 5 + 4 \cdot \left( \frac 12 \right) \\ FG^2 &= 7 \\ FG &= \sqrt 7. \end{align*}

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

Note: The sum of the interior angles of any polygon with $n$ sides is given by $180 ^{\circ} (n - 2)$. Therefore, the sum of the interior angles of a hexagon is $720 ^{\circ}$, and each interior angle of a regular hexagon measures $\frac{720 ^{\circ}}{6} = 120 ^{\circ}$.

Solution 3

Image caption

We use a coordinates approach. Letting the origin be the center of the hexagon, we can let $A = (-1, \sqrt{3}), B = (1, \sqrt{3}), C = (2, 0), D = (1, -\sqrt{3}), E = (-1, -\sqrt{3}), F = (-2, 0).$ Then, $G = (0, \sqrt{3})$ and $H = (0, -\sqrt{3}).$

We use the distance formula four times to get $CH, HF, FG, \text{ and } GC.$

\[CH^2 = (2-0)^2 + (0-(-\sqrt{3}))^2 = 7 \rightarrow CH = \sqrt{7}\] \[HF^2 = (0-(-2))^2 + (-\sqrt{3}-0)^2 = 7 \rightarrow HF = \sqrt{7}\] \[FG^2 = (-2-0)^2 + (0-\sqrt{3})^2 = 7 \rightarrow FG = \sqrt{7}\] \[GC^2 = (0-2)^2 + (\sqrt{3}-0)^2 = 7 \rightarrow GC = \sqrt{7}\]

Thus, the perimeter of $GCHF = CH + HF + FG + GC = \sqrt{7} + \sqrt{7} + \sqrt{7} + \sqrt{7} = \boxed{\textbf{(D)} \ 4\sqrt{7}}$.

~sirswagger21

Note: the last part of this solution could have been simplified by noting that $CH = HF = FG = GC.$

Video Solution 1

https://youtu.be/pIdM5l3CyUY

~Education, the Study of Everything

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

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