Difference between revisions of "2015 AMC 8 Problems/Problem 12"
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===Solution 2=== | ===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. | 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. | ||
+ | -NoisedHens | ||
===Solution 3=== | ===Solution 3=== | ||
− | We can use the feature of 3-Dimension in a cube to solve the problem systematically. 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 | + | We can use the feature of 3-Dimension in a cube to solve the problem systematically. 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}*3 =\boxed{\textbf{(C) } 18}</math>. -----LarryFlora |
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===Video Solution=== | ===Video Solution=== | ||
https://youtu.be/Zhsb5lv6jCI?t=1306 | https://youtu.be/Zhsb5lv6jCI?t=1306 |
Revision as of 13:52, 6 July 2021
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. -NoisedHens
Solution 3
We can use the feature of 3-Dimension in a cube to solve the problem systematically. 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
https://youtu.be/Zhsb5lv6jCI?t=1306
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.