Parallels and Parallelograms

This was an external assessment I had to complete for my IB Mathematics SL course in high school that I recently found again and thought I’d share with everyone. I have no idea what grade I ended up getting on this.

Parallelograms Formed Using Two Parallel Lines and n Transversals

Figure 1 shows that 6 parallelograms are formed when four transversals are present intersecting two parallel lines, with a dotted, lighter line representing a transversal that is ‘overlooked’ when counting the parallelograms.

figure-1

Table 1 shows how many parallelograms are formed given a number of transversals n that intersect two parallel lines using up to 10 transversals.

| Number of Transversals | Parallelograms Formed |
|------------------------|-----------------------|
| 1                      | 0                     |
| 2                      | 1                     |
| 3                      | 3                     |
| 4                      | 6                     |
| 5                      | 10                    |
| 6                      | 15                    |
| 7                      | 21                    |
| 8                      | 28                    |
| 9                      | 36                    |
| 10                     | 45                    |

The equation P=(1/2)n^2-(1/2)n will represent this data perfectly where P represents the number of parallelograms formed using x transversals. The next number adds the number it added to the previous number increment by 1 using the sequence 1, 2, 3, 4, 5, 6, 7, and so on.

0 + 1 = 1, 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15, 15 + 6 = 21, 21 + 7 = 28, and so on.

Parallelograms Formed Using m Parallel Lines with n Transversals

Table 2 shows the number of parallelograms formed by adding both transversals and parallel lines, with the number of parallel lines going down on the left and the number of transversals intersecting those lines going across on the top, up to 10 parallel lines and transversals, starting at 2 parallel lines and 2 transversals since no parallelograms can be formed using only 1 parallel line or only one transversal.

|      || 2    | 3    | 4    | 5    | 6    | 7    | 8    | 9    | 10   |
|------||------|------|------|------|------|------|------|------|------|
| 2    || 1    | 3    | 6    | 10   | 15   | 21   | 28   | 36   | 45   |
| 3    || 3    | 9    | 18   | 30   | 45   | 63   | 84   | 108  | 135  |
| 4    || 6    | 18   | 36   | 60   | 90   | 126  | 168  | 216  | 270  |
| 5    || 10   | 30   | 60   | 100  | 150  | 210  | 280  | 360  | 450  |
| 6    || 15   | 45   | 90   | 150  | 225  | 315  | 420  | 540  | 675  |
| 7    || 21   | 63   | 126  | 210  | 315  | 441  | 588  | 756  | 945  |
| 8    || 28   | 84   | 168  | 280  | 420  | 588  | 784  | 1008 | 1260 |
| 9    || 36   | 108  | 216  | 360  | 540  | 756  | 1008 | 1296 | 1620 |
| 10   || 45   | 135  | 270  | 450  | 675  | 945  | 1260 | 1620 | 2025 |

When putting the values into the table, the numbers filling in the table become a multiplication table, making the number of parallelograms formed the product of the number of parallelograms formed from n transversals and 2 parallel lines by 2 transversals and m parallel lines. Therefore, the equation from before can be done twice for each variable (number of parallel lines and number of transversals) then multiplied together to determine the number of parallelograms formed. Thus, a proper equation to represent the above data would be P=((1/2)m^2-(1/2)m)((1/2)n^2-(1/2)n) where P represents the number of parallelograms formed and n represents the number of transversals intersecting m, the number of parallel lines.

Examples

If m is 9 and n is 6 then:

P=((1/2)(9)^2-(1/2)(9))((1/2)(6)^2-(1/2)(6))=(81/2-9/2)(36/2-6/2)=(72/2)(30/2)=2160/4=540

By checking it with the Table 2 – going across the top to 6 transversals and down to 9 parallel lines – the value given is 540, which matches the answer given by solving Equation 1.

If m is 7 and n is 3 then:

P=((1/2)(7)^2-(1/2)(7))((1/2)(3)^2-(1/2)(3))=(49/2-7/2)(9/2-3/2)=(42/2)(6/2)=252/4=63

Again, by checking it with the Table 2 the value given is 63, which matches the answer given by solving Equation 2.

If m is 17 and n is 11 then:

P=((1/2)(17)^2-(1/2)(17))((1/2)(11)^2-(1/2)(11))=(272/2)(110/2)=29920/4=7480

The value given by Equation 3 would be accepted as an accurate answer without having to go through and count every parallelogram based on the answers increasing exponentially given in Table 2.

Scope and Limitations

After testing all variables up to 10 (as displayed in Table 2) and several random variables past 10, the equation appears to work with any number of n transversals intersecting m parallel lines going from 0 to infinite (since 0 of either would always return 0 parallelograms formed). Negative numbers being inputted into the equation would return invalid results because a negative number of transversals and/or parallel lines cannot occur. All values seem to be reasonable as they follow an exponential growth function, which is also reasonable since each new row or column of parallelograms added increases the total number of parallelograms based on the number of parallelograms that existed beforehand.