Difference between revisions of "2005 AMC 8 Problems/Problem 3"

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==Problem==
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== 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 &lt; 5; ++x )
 
for ( int x = 0; x &lt; 5; ++x )
 
{
 
{
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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>
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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==
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== 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==
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== 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 $\overline{BD}$ of square $ABCD$? [asy] defaultpen(linewidth(1)); for ( int x = 0; x &lt; 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]

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

Rotating square $ABCD$ counterclockwise $45^\circ$ so that the line of symmetry $BD$ is a vertical line makes it easier to see that $\boxed{\textbf{(D)}\ 4}$ squares need to be colored to match its corresponding square.

See Also

2005 AMC 8 (ProblemsAnswer KeyResources)
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

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