Difference between revisions of "1985 AJHSME Problems/Problem 11"
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| − | <math>\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{ | + | <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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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>. | 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>. | ||
| − | The only face <math>\text{X}</math> doesn't share an edge with is <math>\text{Y}</math>, which is choice <math>\boxed{\text{ | + | 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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{{AJHSME box|year=1985|num-b=10|num-a=12}} | {{AJHSME box|year=1985|num-b=10|num-a=12}} | ||
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
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| + | {{MAA Notice}} | ||
Latest revision as of 08:25, 10 January 2023
Contents
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 
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]](http://latex.artofproblemsolving.com/d/a/b/dab8416f1514dce5c2c075749bfab240c1e34b68.png)
Solution
To find the face opposite , we can find the faces sharing an edge with , so the only face remaining will be the opposite face.
Clearly, 
and 
share an edge with . Also, the faces , , and 
share a common vertex, therefore shares an edge with . Similarly, the faces 
, , and share a common vertex, so shares an edge with .
The only face doesn't share an edge with is 
, which is choice 
Video Solution
~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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
