Difference between revisions of "2015 AMC 8 Problems/Problem 12"

Latest revision as of 01:32, 2 March 2024

Problem

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$ , 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$",(1,1,1),N); [/asy]$

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

Solutions

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.

Solution 2

Look at any edge, let's say $\overline{AB}$ . There are three ways we can pair $\overline{AB}$ with another edge. $\overline{AB}\text{ and }\overline{EF}$ , $\overline{AB}\text{ and }\overline{HG}$ , and $\overline{AB}\text{ and }\overline{DC}$ . 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 $\frac{36}{2}$ is $\boxed{\textbf{(C) } 18}$ 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, $\overline{AB}$ , $\overline{BC}$ , and $\overline{BF}$ have $4$ different parallel edges respectively. So it gives us the total pairs of parallel lines are $\binom{4}{2}\cdot3 =\boxed{\textbf{(C) } 18}$ .

--LarryFlora

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/iJC0Wqd1ZcU

~Education, the Study of Everything

Video Solution 2

https://youtu.be/7bgsUa62d4g

~savannahsolver

@@ Line 1: / Line 1: @@
+==Problem==
 How many pairs of parallel edges, such as <math>\overline{AB}</math> and <math>\overline{GH}</math> or <math>\overline{EH}</math> and <math>\overline{FG}</math>, does a cube have?
-<math>\textbf{(A) }6 \quad\textbf{(B) }12 \quad\textbf{(C) } 18 \quad\textbf{(D) } 24 \quad \textbf{(E) } 36</math>
 <asy> import three;
 currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */
@@ Line 18: / Line 18: @@
 label("$F$",(1,1,1),N);
 </asy>
+<math>\textbf{(A) }6\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad \textbf{(E) }36</math>
+==Solutions==
+===Solution 1===
+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.
+===Solution 2===
+Look at any edge, let's say <math>\overline{AB}</math>. There are three ways we can pair <math>\overline{AB}</math> with another edge. <math>\overline{AB}\text{ and }\overline{EF}</math>, <math>\overline{AB}\text{ and }\overline{HG}</math>, and <math>\overline{AB}\text{ and }\overline{DC}</math>. 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 <math>\frac{36}{2}</math> is <math>\boxed{\textbf{(C) } 18}</math> 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, <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==
+{{AMC8 box|year=2015|num-b=11|num-a=13}}
+{{MAA Notice}}

2015 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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