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2023 AMC 8 Problems/Problem 2: Difference between revisions

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→Problem: I have placed asy code for the answers. Could someone please crop the main image until I am done with the paper folding&cutting diagram?
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When a square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line.  When unfolded, the paper will match which of the following figures?
When a square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line.  When unfolded, the paper will match which of the following figures?
[[Image:2023 AMC 8-2.png|thumb|center|800px]]
[[Image:2023 AMC 8-2.png|thumb|center|800px]]
<math>\mathbf{(A)}</math>
<asy>
size(3cm);
path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle;
filldraw(sqA,mediumgrey,black);
</asy>


<math>\mathbf{(B)}</math>
<asy>
size(3cm);
path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle;
filldraw(sqB,mediumgrey,black);
</asy>
<math>\mathbf{(C)}</math><asy>
size(3cm);
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle;
filldraw(sq,mediumgrey,black);
filldraw(sqC,white,black);
</asy>
<math>\mathbf{(D)}</math><asy>
size(3cm);
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle;
filldraw(sq,mediumgrey,black);
filldraw(trD,white,black);
</asy>
<math>\mathbf{(E)}</math><asy>
size(3cm);
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle;
filldraw(sq,mediumgrey,black);
filldraw(sqE,white,black);
</asy>
==Solution 1 (Vague)==
==Solution 1 (Vague)==


Revision as of 18:28, 25 January 2023

Problem

When a square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?

Error creating thumbnail: File missing

$\mathbf{(A)}$ [asy] size(3cm); path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; filldraw(sqA,mediumgrey,black); [/asy]

$\mathbf{(B)}$ [asy] size(3cm); path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; filldraw(sqB,mediumgrey,black); [/asy]

$\mathbf{(C)}$[asy] size(3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; filldraw(sq,mediumgrey,black); filldraw(sqC,white,black); [/asy]

$\mathbf{(D)}$[asy] size(3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; filldraw(sq,mediumgrey,black); filldraw(trD,white,black); [/asy]

$\mathbf{(E)}$[asy] size(3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; filldraw(sq,mediumgrey,black); filldraw(sqE,white,black); [/asy]

Solution 1 (Vague)

Notice how the paper is folded. The bottom right corner of the twice-folded paper has to be the middle of the unfolded paper. So if you cut it in the way that it is shown in the problem, you find (it has to be symmetrical) that the cuts make an equilateral rhombus [tilted square] centered in the middle of the paper.

-claregu

Solution 2 (Thorough)

Notice that when we unfold the paper from the vertical fold line, we get

Error creating thumbnail: File missing

Then, if we unfold the paper from the horizontal fold line, we result in

Error creating thumbnail: File missing

It is clear that the answer is $\boxed{\textbf{(E)}}$

~MrThinker

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=5658

See Also

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
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

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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