2021 AMC 12B Problems/Problem 15

The following problem is from both the 2021 AMC 10B #20 and 2021 AMC 12B #15, so both problems redirect to this page.

1 Problem
2 Solution 1
3 Solution 2
4 Video Solution (🚀Under 3 min!🚀)
5 Video Solution by OmegaLearn (Extending Lines, Angle Chasing, Trig Area)
6 Video Solution by Hawk Math
7 Video Solution by Mathematical Dexterity (Basic Area Formulas)
8 Video Solution by TheBeautyofMath
9 Video Solution by Interstigation (Ignore Useless Segments)
10 Video Solution by The Power of Logic
11 Remark
12 See Also

Problem

The figure is constructed from $11$ line segments, each of which has length $2$ . The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$ , where $m$ and $n$ are positive integers. What is $m + n ?$ $[asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,S); label("D",D,S); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy]$

$\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24$

Solution 1

Let $M$ be the midpoint of $CD$ . Noting that $AED$ and $ABC$ are $120$ - $30$ - $30$ triangles because of the equilateral triangles, $AM=\sqrt{AD^2-MD^2}=\sqrt{12-1}=\sqrt{11} \implies [ACD]=\sqrt{11}.$ Also, $[AED]=2\cdot2\cdot\frac{1}{2}\cdot\sin{120^\circ}=\sqrt{3}$ (or split $\triangle AED$ into two $30-60-90$ triangles) and so $[ABCDE]=[ACD]+2[AED]=\sqrt{11}+2\sqrt{3}=\sqrt{11}+\sqrt{12} \implies \boxed{\textbf{(D)} ~23}.$

Solution 2

$[asy] /* Made by samrocksnature, adapted by Tucker, then adjusted by samrocksnature again, then adjusted by erics118 xD*/ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-1.85,2); pair G=(-3.1,2); draw(A--G--A--F, lightgray); draw(B--F--C, lightgray); draw(E--G--D, lightgray); dot(F^^G, lightgray); draw(A--B--C--D--E--A); draw(A--C--A--D); label("A",A,N); label("B",B,W); label("C",C,S); label("D",D,S); label("E",E,dir(0)); dot(A^^B^^C^^D^^E); [/asy]$

Draw diagonals $AC$ and $AD$ to split the pentagon into three parts. We can compute the area for each triangle and sum them up at the end. For triangles $ABC$ and $ADE$ , they each have area $2\cdot\frac{1}{2}\cdot\frac{4\sqrt{3}}{4}=\sqrt{3}$ . For triangle $ACD$ , we can see that $AC=AD=2\sqrt{3}$ and $CD=2$ . Using Pythagorean Theorem, the altitude for this triangle is $\sqrt{11}$ , so the area is $\sqrt{11}$ . Adding each part up, we get $2\sqrt{3}+\sqrt{11}=\sqrt{12}+\sqrt{11} \implies \boxed{\textbf{(D)} ~23}$ .

Video Solution (🚀Under 3 min!🚀)

https://youtu.be/1CAbbfArA6w

~Education, the Study of Everything

Video Solution by OmegaLearn (Extending Lines, Angle Chasing, Trig Area)

https://youtu.be/QtSbAKUb1VE

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=p4iCAZRUESs

Video Solution by Mathematical Dexterity (Basic Area Formulas)

https://www.youtube.com/watch?v=7kDTlVixsD0

Video Solution by TheBeautyofMath

https://youtu.be/FV9AnyERgJQ?t=1226

~IceMatrix

Video Solution by Interstigation (Ignore Useless Segments)

https://youtu.be/9eChInz03UQ

~Interstigation

Video Solution by The Power of Logic

https://www.youtube.com/watch?v=f8L5K2yIXUc

~The Power of Logic

Remark

This configuration of $11$ congruent line segments is known as the Moser Spindle https://en.wikipedia.org/wiki/Moser_spindle , and can be used to demonstrate that $3$ colors are not sufficient to color all of the points in the plane such that points that are $1$ unit apart have different colors. Finding the minimum such number of colors is a famous unsolved problem: the Nelson-Hadwiger problem. See: https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem

~hailstone

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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 10 Problems and Solutions

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