AMC 8 2015 Problem 12
AMC8 2015 年第 12 题
立体几何★★★☆☆
How many pairs of parallel edges, such as , and , or , and , does a cube have? Repeating parallel edges do not count toward the final answer.
在一个正方体中，有多少对平行的棱（例如 和 ，或者 和 ）？重复的平行棱不计入最终答案。
- A.6
- B.12
- C.18
- D.24
- E.36
正确答案 · Correct Answer
C
解析 · Solution
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 6 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.
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 18 total pairs of parallel lines.
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 4 different parallel edges respectively. So it gives us the total pairs of parallel lines are .
Our first case are lines on the same plane. There are two pairs of parallel lines for a square, and there are 6 squares, which give us cases. The next cases we need to analyze are the parallel lines on opposite planes (lines such as ). There are 2 pairs for every 2 planes, and there are 6 planes in total, giving us cases. Adding up the cases, we get 12+6 = 18
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