Difference between revisions of "2001 AMC 12 Problems/Problem 10"
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== Problem == | == Problem == | ||
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The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to | The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to | ||
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<asy> | <asy> | ||
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} | } | ||
</asy> | </asy> | ||
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| + | <math> \textbf{(A) }50 \qquad \textbf{(B) }52 \qquad \textbf{(C) }54 \qquad \textbf{(D) }56 \qquad \textbf{(E) }58 </math> | ||
== Solution == | == Solution == | ||
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Consider any single tile: | Consider any single tile: | ||
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</asy> | </asy> | ||
| − | If the side of the small square is <math>a</math>, then the area of the tile is <math>9a^2</math>, with <math>4a^2</math> covered by squares and <math>5a^2</math> by pentagons. | + | If the side of the small square is <math>a</math>, then the area of the tile is <math>\left(3a\right)^2 = 9a^2</math>, with <math>4a^2</math> covered by squares, and therefore <math>9a^2-4a^2 = 5a^2</math> by pentagons. |
| − | Hence exactly <math>5 | + | Hence exactly <math>\frac{5}{9}</math> of any tile is covered by pentagons, and therefore pentagons cover <math>\frac{5}{9}</math> of the plane. When expressed as a percentage, this is <math>55.\overline{5}\%</math>, and the closest integer to this value is <math>\boxed{\textbf{(D) }56}</math>. |
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| + | == Video Solution == | ||
| + | https://www.youtube.com/watch?v=P43NYTbZH4k | ||
== See Also == | == See Also == | ||
Latest revision as of 05:25, 6 June 2025
- The following problem is from both the 2001 AMC 12 #10 and 2001 AMC 10 #18, so both problems redirect to this page.
Contents
Problem
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
![[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]](http://latex.artofproblemsolving.com/2/c/8/2c8743395e4a086d2ae9a30a2d161157e7dc3077.png)
Solution
Consider any single tile:
![[asy] unitsize(1cm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; draw(p); [/asy]](http://latex.artofproblemsolving.com/3/6/f/36f7a20120732a8955c3f77ea0d1e81fc34cc6ea.png)
If the side of the small square is 
, then the area of the tile is 
, with 
covered by squares, and therefore 
by pentagons.
Hence exactly 
of any tile is covered by pentagons, and therefore pentagons cover of the plane. When expressed as a percentage, this is 
, and the closest integer to this value is 
.
Video Solution
https://www.youtube.com/watch?v=P43NYTbZH4k
See Also
| 2001 AMC 12 (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 | |
| 2001 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
