2023 AMC 8 Problems/Problem 2

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?

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

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

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

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

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

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

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

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

~MrThinker

Video Solution by Magic Square

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

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

Page

Toolbox

Search

2023 AMC 8 Problems/Problem 2

Contents

Problem

Solution 1 (Vague)

Solution 2 (Thorough)

Video Solution by Magic Square

See Also