Difference between revisions of "2021 Fall AMC 10B Problems/Problem 2"
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==Solution #3 (Overkill)== | ==Solution #3 (Overkill)== | ||
| − | We start by finding the points. The outlined shape is made up of <math>(1,0),(3,5),(5,0),(3,2)</math>. By the | + | We start by finding the points. The outlined shape is made up of <math>(1,0),(3,5),(5,0),(3,2)</math>. By the https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem, we find the area to be <math>6</math>, or <math>\boxed{B}</math>. |
~Taco12 | ~Taco12 | ||
| − | |||
==Solution #4 (Pick's Theorem)== | ==Solution #4 (Pick's Theorem)== | ||
Revision as of 14:06, 27 November 2021
Contents
Problem
What is the area of the shaded figure shown below?
![[asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.15,0), EndArrow); draw(O--Y+(0,0.15), EndArrow); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]](http://latex.artofproblemsolving.com/1/7/1/1714dc248f12bd0640e9ec37023a14eddda46d9c.png)
Solution #1
We have 
isosceles triangles. Thus, the area of the shaded region is 
Thus our answer is 
~NH14
Solution #2
As we can see, the shape is symmetrical, so it will be equally valid to simply calculate one of the half's area and multiply by 2. One half's area is 
, so two halves would be 
. Thus our answer is
~Hefei417, or 陆畅 Sunny from China
Solution #3 (Overkill)
We start by finding the points. The outlined shape is made up of 
. By the https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem, we find the area to be 
, or 
.
~Taco12
Solution #4 (Pick's Theorem)
We can use Pick's Theorem. We have 
interior points and boundary points. By Pick's Theorem, we get 
Checking our answer choices, we find our answer to be .
~danprathab
Solution 5
The area is
Therefore, the answer is 
.
~Steven Chen (www.professorchenedu.com)
Video Solution by Interstigation
https://youtu.be/p9_RH4s-kBA?t=110
See Also
| 2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
