# Mathematics SL Parellels and Parallelograms. This task will consider the number of parallelograms formed by intersecting m horizontal parallel lines with n parallel transversals; we are to deduce a formula that will satisfy the above.

Jia-Der Ju Wang

Parallels and Parallelograms

Mathematics Coursework

This task will consider the number of parallelograms formed by intersecting m horizontal parallel lines with n parallel transversals; we are to deduce a formula that will satisfy the above.

Methodology

1. We started out the investigation with a pair of horizontal parallel lines and a pair of parallel transversals. One parallelogram (A1) is formed (shown in Figure 1)

1. A third parallel transversal is added to the diagram as shown in Figure 2. Three parallelograms are formed: A1, A2, and A1A2

1. When a fourth transversal is added to Figure 2 (Figure 3), six parallelograms are formed. A1, A2, A3, A1A2, A2A3, A1A3

1. Figure 4 has 5 transversals cutting the pair of horizontal parallels, forming ten parallelograms. A1, A2, A3, A4, A1A2, A2A3, A3A4, A1A3, A2A4, A1A4

1. A sixth transversal was added to Figure 5, forming 15 parallelograms shown in Figure 6. A1, A2, A3, A4, A5, A1A2, A2A3, A3A4, A4A5, A1A3, A2A4, A3A5, A1A4, A2A5, A1A5

1. When a seventh transversal is added, twenty-one parallelograms are formed (Figure 7). A1, A2, A3, A4, A5, A6, A1A2, A2A3, A3A4, A4A5, A5A6, A1A3, A2A4, A3A5, A4A6, A1A4, A2A5, A3A6, A1A5, A2A6, A1A6.

1. I then used technology (table 1.0) to record the above and calculate the differences between the parallelograms formed with each addition of a transversal.

Table 1.0

At this point, I realized that the ‘First difference between terms’ was somewhat similar to what happens with the terms in a triangle sequence:

Also since the number of parallelograms created as the number of transversals increased each had a Second Order difference of 1, it was immediately known that the general formula must be a quadratic equation. So to find the formula I followed the following steps:

Then solve them as simultaneous equations

...