Difference between revisions of "2015 AMC 8 Problems/Problem 12"
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+ | ==Problem== | ||
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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? | 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? | ||
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<asy> import three; | <asy> import three; | ||
currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ | currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ | ||
Line 18: | Line 18: | ||
label("$F$",(1,1,1),N); | label("$F$",(1,1,1),N); | ||
</asy> | </asy> | ||
− | ==Solution== | + | |
+ | <math>\text{(A) }6 \quad\text{(B) }12 \quad\text{(C) } 18 \quad\text{(D) } 24 \quad \text{(E) } 36</math> | ||
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+ | |||
+ | ==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. | 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=== | ||
+ | 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>. | ||
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+ | --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 15:47, 14 May 2023
Contents
Problem
How many pairs of parallel edges, such as and or and , does a cube have?
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 |
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