Difference between revisions of "2015 AMC 8 Problems/Problem 12"
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| − | <math>\ | + | <math>\textbf{(A) }6\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad \textbf{(E) }36</math> |
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==Solutions== | ==Solutions== | ||
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===Solution 3=== | ===Solution 3=== | ||
| − | We can use the feature of 3-Dimension in a cube to solve the problem systematically. | + | We can use the feature of 3-Dimension in a cube to solve the problem systematically. For example, in the 3-D of the cube, <math>\overline{AB}</math>, <math>\overline{BC}</math>, and <math>\overline{BF}</math> have <math>4</math> different parallel edges respectively. So it gives us the total pairs of parallel lines are <math>\binom{4}{2}\cdot3 =\boxed{\textbf{(C) } 18}</math>. |
| − | + | ||
| − | + | --LarryFlora | |
| + | |||
| + | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | ||
| + | https://youtu.be/iJC0Wqd1ZcU | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution 2== | ||
| + | https://youtu.be/7bgsUa62d4g | ||
| − | + | ~savannahsolver | |
| − | |||
==See Also== | ==See Also== | ||
Latest revision as of 01:32, 2 March 2024
Contents
Problem
How many pairs of parallel edges, such as 
and 
or 
and 
, does a cube have?
![[asy] import three; currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1)); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); label("$D$",(0,0,0),S); label("$A$",(0,0,1),N); label("$H$",(0,1,0),S); label("$E$",(0,1,1),N); label("$C$",(1,0,0),S); label("$B$",(1,0,1),N); label("$G$",(1,1,0),S); label("$F$",(1,1,1),N); [/asy]](http://latex.artofproblemsolving.com/f/1/b/f1b4f8e43db0ad2a38957a7e452bb8c8688d5906.png)
Solutions
Solution 1
We first count the number of pairs of parallel lines that are in the same direction as . The pairs of parallel lines are 
, 
, 
, 
, 
, and 
. These are 
pairs total. We can do the same for the lines in the same direction as 
and 
. This means there are 
total pairs of parallel lines.
Solution 2
Look at any edge, let's say . There are three ways we can pair with another edge. , 
, and 
. There are 12 edges on a cube. 3 times 12 is 36. We have to divide by 2 because every pair is counted twice, so 
is 
total pairs of parallel lines.
Solution 3
We can use the feature of 3-Dimension in a cube to solve the problem systematically. For example, in the 3-D of the cube, , 
, and 
have 
different parallel edges respectively. So it gives us the total pairs of parallel lines are 
.
--LarryFlora
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution 2
~savannahsolver
See Also
| 2015 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 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. 
