Parallelogram Formation from Parallel Lines

To determine the total number of parallelograms formed when two sets of parallel lines intersect, we need to understand the definition of a parallelogram and use combinatorial principles, specifically combinations.

Understanding Parallelogram Requirements

A parallelogram is a quadrilateral (a four-sided polygon) where opposite sides are parallel. When two distinct sets of parallel lines intersect, any choice of two lines from the first set and two lines from the second set will automatically form a parallelogram. The intersection points of these four chosen lines define the vertices of the parallelogram.

• We are given a first set consisting of four parallel lines.
• We are also given a second set consisting of four other parallel lines, which intersect the lines of the first set.

Calculating Parallelograms Using Combinations

To form a parallelogram, we must select two lines from the first set of parallel lines and two lines from the second set of parallel lines. The order in which we select the lines does not matter, which indicates that this is a problem solvable with combinations.

The formula for combinations is given by:

$$C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}$$

where $n$ is the total number of items to choose from, and $k$ is the number of items to choose.

Step-by-Step Calculation of Parallelograms:

1. Choosing lines from the first set:

   From the first set of four parallel lines, we need to choose two lines to form two sides of the parallelogram.

   Using the combination formula with $n=4$ (total lines in the first set) and $k=2$ (lines to choose):

   $$C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{(4 \times 3 \times 2 \times 1)}{(2 \times 1)(2 \times 1)} = \frac{24}{4} = 6$$

   So, there are 6 ways to select two lines from the first set.

2. Choosing lines from the second set:

   Similarly, from the second set of four other parallel lines, we need to choose two lines to form the remaining two sides of the parallelogram.

   Using the combination formula again with $n=4$ (total lines in the second set) and $k=2$ (lines to choose):

   $$C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{(4 \times 3 \times 2 \times 1)}{(2 \times 1)(2 \times 1)} = \frac{24}{4} = 6$$

   Thus, there are also 6 ways to select two lines from the second set.

3. Total number of parallelograms:

   To find the total number of parallelograms that can be formed, we multiply the number of ways to choose lines from the first set by the number of ways to choose lines from the second set. This is because each combination of two lines from the first set can be paired with each combination of two lines from the second set to form a unique parallelogram.

   Total Parallelograms = (Number of ways to choose from Set 1) $\times$ (Number of ways to choose from Set 2)

   Total Parallelograms = $C(4, 2) \times C(4, 2) = 6 \times 6 = 36$

Therefore, a total of 36 parallelograms can be formed by the intersection of four parallel lines with four other parallel lines.