Difference between revisions of "2005 AMC 8 Problems/Problem 3"
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| − | ==Problem== | + | == Problem == |
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal <math> \overline{BD}</math> of square <math> ABCD</math>? | What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal <math> \overline{BD}</math> of square <math> ABCD</math>? | ||
| − | <asy>defaultpen(linewidth(1)); | + | <asy> |
| + | defaultpen(linewidth(1)); | ||
for ( int x = 0; x < 5; ++x ) | for ( int x = 0; x < 5; ++x ) | ||
{ | { | ||
| Line 15: | Line 16: | ||
label("$B$", (4, 4), NE); | label("$B$", (4, 4), NE); | ||
label("$C$", (4, 0), SE); | label("$C$", (4, 0), SE); | ||
| − | label("$D$", (0, 0), SW);</asy> | + | label("$D$", (0, 0), SW); |
| + | </asy> | ||
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math> | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math> | ||
| − | ==Solution== | + | == Solution == |
Rotating square <math>ABCD</math> counterclockwise <math>45^\circ</math> so that the line of symmetry <math>BD</math> is a vertical line makes it easier to see that <math>\boxed{\textbf{(D)}\ 4}</math> squares need to be colored to match its corresponding square. | Rotating square <math>ABCD</math> counterclockwise <math>45^\circ</math> so that the line of symmetry <math>BD</math> is a vertical line makes it easier to see that <math>\boxed{\textbf{(D)}\ 4}</math> squares need to be colored to match its corresponding square. | ||
| − | ==See Also== | + | == See Also == |
{{AMC8 box|year=2005|num-b=2|num-a=4}} | {{AMC8 box|year=2005|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 12:33, 19 October 2020
Problem
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal 
of square 
?
![[asy] defaultpen(linewidth(1)); for ( int x = 0; x < 5; ++x ) { draw((0,x)--(4,x)); draw((x,0)--(x,4)); } fill((1,0)--(2,0)--(2,1)--(1,1)--cycle); fill((0,3)--(1,3)--(1,4)--(0,4)--cycle); fill((2,3)--(4,3)--(4,4)--(2,4)--cycle); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle); label("$A$", (0, 4), NW); label("$B$", (4, 4), NE); label("$C$", (4, 0), SE); label("$D$", (0, 0), SW); [/asy]](http://latex.artofproblemsolving.com/7/5/f/75f8ad6c87a6d20912c9a932a8e8dbc6b6364d2d.png)
Solution
Rotating square counterclockwise 
so that the line of symmetry 
is a vertical line makes it easier to see that 
squares need to be colored to match its corresponding square.
See Also
| 2005 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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. 
