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Difference between revisions of "2015 AMC 8 Problems/Problem 12"
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We first count the number of pairs of parallel lines that are in the same direction as <math>\overline{AB}</math>.  The pairs of parallel lines are <math>\overline{AB}\text{ and }\overline{EF}</math>, <math>\overline{CD}\text{ and }\overline{GH}</math>, <math>\overline{AB}\text{ and }\overline{CD}</math>, <math>\overline{EF}\text{ and }\overline{GH}</math>, <math>\overline{AB}\text{ and }\overline{GH}</math>, and <math>\overline{CD}\text{ and }\overline{EF}</math>.  These are <math>6</math> pairs total.  We can do the same for the lines in the same direction as <math>\overline{AE}</math> and <math>\overline{AD}</math>.  This means there are <math>6\cdot 3=\boxed{\textbf{(C) } 18}</math> total pairs of parallel lines.
 
We first count the number of pairs of parallel lines that are in the same direction as <math>\overline{AB}</math>.  The pairs of parallel lines are <math>\overline{AB}\text{ and }\overline{EF}</math>, <math>\overline{CD}\text{ and }\overline{GH}</math>, <math>\overline{AB}\text{ and }\overline{CD}</math>, <math>\overline{EF}\text{ and }\overline{GH}</math>, <math>\overline{AB}\text{ and }\overline{GH}</math>, and <math>\overline{CD}\text{ and }\overline{EF}</math>.  These are <math>6</math> pairs total.  We can do the same for the lines in the same direction as <math>\overline{AE}</math> and <math>\overline{AD}</math>.  This means there are <math>6\cdot 3=\boxed{\textbf{(C) } 18}</math> total pairs of parallel lines.
  
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==Solution 2==
 
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Pick a random edge. Given another edge, the probability that it is parallel to this edge is <math>\frac{3}{12-1}=\frac{3}{11}</math>. Keep in mind we already used one edge. There are <math>12</math> edges so <math>\binom{12}{2}=66</math> pairs. So our answer is <math>\frac{3}{11} \times 66=\boxed{\textbf{(C)}~18}</math>.
 
 
==See Also==
 
==See Also==
  
 
{{AMC8 box|year=2015|num-b=11|num-a=13}}
 
{{AMC8 box|year=2015|num-b=11|num-a=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 17:35, 10 September 2017

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?


$\textbf{(A) }6 \quad\textbf{(B) }12 \quad\textbf{(C) } 18 \quad\textbf{(D) } 24 \quad \textbf{(E) } 36$ [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]

Solution 1

We first count the number of pairs of parallel lines that are in the same direction as $\overline{AB}$. The pairs of parallel lines are $\overline{AB}\text{ and }\overline{EF}$, $\overline{CD}\text{ and }\overline{GH}$, $\overline{AB}\text{ and }\overline{CD}$, $\overline{EF}\text{ and }\overline{GH}$, $\overline{AB}\text{ and }\overline{GH}$, and $\overline{CD}\text{ and }\overline{EF}$. These are $6$ pairs total. We can do the same for the lines in the same direction as $\overline{AE}$ and $\overline{AD}$. This means there are $6\cdot 3=\boxed{\textbf{(C) } 18}$ total pairs of parallel lines.

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

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