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Difference between revisions of "1985 AJHSME Problems/Problem 11"

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</asy>
 
</asy>
  
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<math>\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{Y} \qquad \text{(E)}\ \text{Z}</math>
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<math>\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{W} \qquad \text{(E)}\ \text{Y}</math>
  
 
==Solution==
 
==Solution==
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To find the face opposite <math>\text{X}</math>, we can find the faces sharing an edge with <math>\text{X}</math>, so the only face remaining will be the opposite face.
 
To find the face opposite <math>\text{X}</math>, we can find the faces sharing an edge with <math>\text{X}</math>, so the only face remaining will be the opposite face.
  
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Clearly, <math>\text{V}</math> and <math>\text{Z}</math> share an edge with <math>\text{X}</math>.  Also, the faces <math>\text{V}</math>, <math>\text{X}</math>, and <math>\text{W}</math> share a common vertex, therefore <math>\text{X}</math> shares an edge with <math>\text{W}</math>.  Similarly, the faces <math>\text{U}</math>, <math>\text{V}</math>, and <math>\text{X}</math> share a common vertex, so <math>\text{X}</math> shares an edge with <math>\text{W}</math>.   
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Clearly, <math>\text{V}</math> and <math>\text{Z}</math> share an edge with <math>\text{X}</math>.  Also, the faces <math>\text{V}</math>, <math>\text{X}</math>, and <math>\text{W}</math> share a common vertex, therefore <math>\text{X}</math> shares an edge with <math>\text{W}</math>.  Similarly, the faces <math>\text{U}</math>, <math>\text{V}</math>, and <math>\text{X}</math> share a common vertex, so <math>\text{X}</math> shares an edge with <math>\text{U}</math>.   
  
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The only face <math>\text{X}</math> doesn't share an edge with is <math>\text{Y}</math>, which is choice <math>\boxed{\text{D}}</math>
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The only face <math>\text{X}</math> doesn't share an edge with is <math>\text{Y}</math>, which is choice <math>\boxed{\text{E}}</math>
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 +
==Video Solution==
 +
https://youtu.be/YE9u_O5d-Eg
 +
 
 +
~savannahsolver
  
 
==See Also==
 
==See Also==
  
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[[1985 AJHSME Problems]]
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{{AJHSME box|year=1985|num-b=10|num-a=12}}
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[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 08:25, 10 January 2023

Problem

A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\text{X}$ is

[asy] draw((0,0)--(0,1)--(2,1)--(2,2)--(3,2)--(3,0)--(2,0)--(2,-2)--(1,-2)--(1,0)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((1,0)--(2,0)); draw((1,-1)--(2,-1)); draw((2,1)--(3,1)); label("U",(.5,.3),N); label("V",(1.5,.3),N); label("W",(2.5,.3),N); label("X",(1.5,-.7),N); label("Y",(2.5,1.3),N); label("Z",(1.5,-1.7),N); [/asy]

$\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{W} \qquad \text{(E)}\ \text{Y}$

Solution

To find the face opposite $\text{X}$, we can find the faces sharing an edge with $\text{X}$, so the only face remaining will be the opposite face.

Clearly, $\text{V}$ and $\text{Z}$ share an edge with $\text{X}$. Also, the faces $\text{V}$, $\text{X}$, and $\text{W}$ share a common vertex, therefore $\text{X}$ shares an edge with $\text{W}$. Similarly, the faces $\text{U}$, $\text{V}$, and $\text{X}$ share a common vertex, so $\text{X}$ shares an edge with $\text{U}$.

The only face $\text{X}$ doesn't share an edge with is $\text{Y}$, which is choice $\boxed{\text{E}}$

Video Solution

https://youtu.be/YE9u_O5d-Eg

~savannahsolver

See Also

1985 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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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