Difference between revisions of "2001 AMC 10 Problems/Problem 5"

(Problem)
 
(11 intermediate revisions by 7 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
  
How many of the twelve pentominoes at least one line of symmetry?
+
How many of the twelve pentominoes pictured below have at least one line of reflectional 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 ==
 +
 +
[[File:Pentonimoes.gif]]
 +
 +
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.
 +
 +
==Video Solution by Daily Dose of Math==
 +
 +
https://youtu.be/svFpNvUUY7E?si=CloMWtqbbhBNgWy_
 +
 +
~Thesmartgreekmathdude
 +
 +
== See Also ==
 +
 +
{{AMC10 box|year=2001|num-b=4|num-a=6}}
 +
{{MAA Notice}}

Latest revision as of 15:11, 15 July 2024

Problem

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

Pentonimoes.gif

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 (ProblemsAnswer KeyResources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png