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2001 AMC 10 Problems/Problem 5: Difference between revisions

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How many of the twelve pentominoes pictured below have at least one line of symmetry?
How many of the twelve pentominoes pictured below have at least one line of symmetry?


<math> \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7 </math>
<asy>
unitsize(5mm);
defaultpen(linewidth(1pt));
draw(shift(2,0)*unitsquare);
draw(shift(2,1)*unitsquare);
draw(shift(2,2)*unitsquare);
draw(shift(1,2)*unitsquare);
draw(shift(0,2)*unitsquare);
draw(shift(2,4)*unitsquare);
draw(shift(2,5)*unitsquare);
draw(shift(2,6)*unitsquare);
draw(shift(1,5)*unitsquare);
draw(shift(0,5)*unitsquare);
draw(shift(4,8)*unitsquare);
draw(shift(3,8)*unitsquare);
draw(shift(2,8)*unitsquare);
draw(shift(1,8)*unitsquare);
draw(shift(0,8)*unitsquare);
draw(shift(6,8)*unitsquare);
draw(shift(7,8)*unitsquare);
draw(shift(8,8)*unitsquare);
draw(shift(9,8)*unitsquare);
draw(shift(9,9)*unitsquare);
draw(shift(6,5)*unitsquare);
draw(shift(7,5)*unitsquare);
draw(shift(8,5)*unitsquare);
draw(shift(7,6)*unitsquare);
draw(shift(7,4)*unitsquare);
draw(shift(6,1)*unitsquare);
draw(shift(7,1)*unitsquare);
draw(shift(8,1)*unitsquare);
draw(shift(6,0)*unitsquare);
draw(shift(7,2)*unitsquare);
draw(shift(11,8)*unitsquare);
draw(shift(12,8)*unitsquare);
draw(shift(13,8)*unitsquare);
draw(shift(14,8)*unitsquare);
draw(shift(13,9)*unitsquare);
draw(shift(11,5)*unitsquare);
draw(shift(12,5)*unitsquare);
draw(shift(13,5)*unitsquare);
draw(shift(11,6)*unitsquare);
draw(shift(13,4)*unitsquare);
draw(shift(11,1)*unitsquare);
draw(shift(12,1)*unitsquare);
draw(shift(13,1)*unitsquare);
draw(shift(13,2)*unitsquare);
draw(shift(14,2)*unitsquare);
draw(shift(16,8)*unitsquare);
draw(shift(17,8)*unitsquare);
draw(shift(18,8)*unitsquare);
draw(shift(17,9)*unitsquare);
draw(shift(18,9)*unitsquare);
draw(shift(16,5)*unitsquare);
draw(shift(17,6)*unitsquare);
draw(shift(18,5)*unitsquare);
draw(shift(16,6)*unitsquare);
draw(shift(18,6)*unitsquare);
draw(shift(16,0)*unitsquare);
draw(shift(17,0)*unitsquare);
draw(shift(17,1)*unitsquare);
draw(shift(18,1)*unitsquare);
draw(shift(18,2)*unitsquare);</asy>
 
<math>\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7</math>


== Solution ==
== Solution ==
Line 11: Line 75:
The ones with lines over the shapes have at least one line of symmetry. Counting the number of shapes that have line(s) on them,
The ones with lines over the shapes have at least one line of symmetry. Counting the number of shapes that have line(s) on them,
we find <math> \boxed{\textbf{(D)}\ 6} </math> pentominoes.
we find <math> \boxed{\textbf{(D)}\ 6} </math> pentominoes.
==Video Solution by Daily Dose of Math==
https://youtu.be/svFpNvUUY7E?si=CloMWtqbbhBNgWy_
~Thesmartgreekmathdude


== See Also ==
== See Also ==
Line 16: Line 86:
{{AMC10 box|year=2001|num-b=4|num-a=6}}
{{AMC10 box|year=2001|num-b=4|num-a=6}}
{{MAA Notice}}
{{MAA Notice}}
[[Category: Introductory Geometry Problems]]

Latest revision as of 16:13, 18 October 2025

Problem

How many of the twelve pentominoes pictured below have at least one line of symmetry?

[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy]

$\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$

Solution

The ones with lines over the shapes have at least one line of symmetry. Counting the number of shapes that have line(s) on them, we find $\boxed{\textbf{(D)}\ 6}$ pentominoes.

Video Solution by Daily Dose of Math

https://youtu.be/svFpNvUUY7E?si=CloMWtqbbhBNgWy_

~Thesmartgreekmathdude

See Also

2001 AMC 10 (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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. Error creating thumbnail: File missing

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