Difference between revisions of "2018 AMC 8 Problems/Problem 4"
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We find <math>14</math> as our answer instead of the correct answer of <math>13</math>. | We find <math>14</math> as our answer instead of the correct answer of <math>13</math>. | ||
| − | Does it actually work or is there a flaw in my reason? | + | Does it actually not work or is there a flaw in my reason? |
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Revision as of 17:24, 17 June 2022
Problem
The twelve-sided figure shown has been drawn on 
graph paper. What is the area of the figure in 
?
![[asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp); [/asy]](http://latex.artofproblemsolving.com/f/4/d/f4d92072ddf7f5314e710dc9641eb84c3ea32879.png)
Solution 1
We count 
unit squares in the middle, and 
small triangles, which gives 4 rectangles each with an area of 
. Thus, the answer is 
Solution 2
We can see here that there are 
total squares in the middle. We also see that the triangles that make the corners of the shape have an area half the squares' area. Then we can easily find that each corner has an area of one square and there are 
corners so we add that to the original 9 squares to get That is how I did it ~avamarora
Note
Notice here that Pick's Theorem DOES NOT WORK.
Reason:
There are lattice points, and 
lattice points on the boundary. Then,
![\[8 + 12 \div 2 = 14\]](http://latex.artofproblemsolving.com/b/d/9/bd948d0237123f10ed62cfca4a3068387dce0fcc.png)
We find 
as our answer instead of the correct answer of 
.
Does it actually not work or is there a flaw in my reason?
Video Solution
~savannahsolver
See Also
| 2018 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
