Question

17
how many pairs of parallel line segments are shown?

18

8

4

24

Explanation:

Step1: Analyze the cube structure

In a cube (or a rectangular prism), each set of parallel edges (line segments) can be grouped by their direction. Let's consider the three dimensions (length, width, height) or the three pairs of opposite faces.

First, identify the types of line segments:

• Let's name the edges: For example, horizontal (front-back), vertical (up-down), and depth (left-right) or based on the cube's faces.

Step2: Count parallel pairs for each direction

1. First set of parallel segments (e.g., \(AB\)-like):

Segments like \(AB\), \(CD\), \(EF\), \(GH\). The number of pairs here: The number of ways to choose 2 from 4 is \(\binom{4}{2}=\frac{4!}{2!(4 - 2)!}=\frac{4\times3}{2\times1}=6\). But wait, actually, in a cube, for each of the three directions (x, y, z), we have 4 edges. Wait, no, looking at the diagram, maybe it's a rectangular prism with some edges. Wait, the diagram has vertices: \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\). Let's list the edges:

• Edges parallel to \(AB\): \(AB\), \(CD\), \(EF\), \(GH\)? Wait, no, \(AB\) is vertical? Wait, maybe the cube has three sets of 4 parallel edges? No, wait, in a cube, each edge has 3 other edges parallel to it? Wait, no, in a rectangular prism (cube is a special case), for each edge, there are 3 edges parallel? No, wait, let's correct:

In a rectangular prism, there are 12 edges, divided into 3 groups of 4 parallel edges (each group corresponding to length, width, height). Wait, no, 12 edges: 4 of length, 4 of width, 4 of height. So for each group of 4 edges (same direction), the number of parallel pairs is \(\binom{4}{2}=\frac{4\times3}{2}=6\). Then, since there are 3 such groups (length, width, height), total pairs would be \(3\times6 = 18\)? Wait, but let's check the diagram.

Wait, the diagram: Let's see the edges. Let's list the edges:

• \(AB\), \(AD\), \(AE\) (wait, no, the diagram: \(A\) connected to \(B\), \(A\) connected to \(E\), \(A\) connected to \(C\)? Wait, maybe it's a cube with some edges. Wait, the problem is about parallel line segments. Let's think again.

Wait, the options include 18. Let's re-express:

In a rectangular prism (the figure here, probably a cube or rectangular box), there are three sets of parallel edges (each set has 4 edges). For each set of 4 edges, the number of parallel pairs is \(\binom{4}{2}=\frac{4!}{2!2!}=6\). So three sets: \(3\times6 = 18\). So that's the calculation.

Answer:

18 (corresponding to the option "18")