# Parallels and Parallelograms Maths Investigation.

by julianan (student)

## Definitions:

Transversal:  that cuts across a set of lines or the  of a . Transversals often cut across .

Parallel line: Two    that do not intersect. Note: Parallel lines have the same .

Parallelogram:  with two pairs of  .

A1 :

ᴗ :

This investigation aims at finding a relationship between the numbers of horizontal parallel lines and the transversals. When these lines intersect they form parallelograms. The aim of this investigation is examine and determine a general statement for transversals and horizontal lines and how they affect the number of parallelograms formed within the figure. A diagram of a parallelogram and a transversal is shown below.

Figure 1: two transversals

Figure 1 below shows a pair of horizontal parallel lines and a pair of parallel transversals. One parallelogram (A1) is formed. A1

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

A2

A1

Figure 2: three transversals

Adding a third transversal gives us a total of three parallelograms.

Figure 3: four transversals

A1 , A2 , A3, A1  A2, A2  A3  and A1  A3.

Adding a fourth transversals gives us a total of six (6) parallelograms.

Figure 4: five transversals

A1 , A2 , A3, A4  ,A1  A2, A1  A3, A1  A4 , A2  A3, A2  A4  A3  A4.

Adding a fifth transversal gives us a total of ten parallelograms.

Figure 5: six transversals

A1 , A2 , A3, A4, A5 , A1  A2, A1  A3, A1  A4 , A1  A5 , A2  A3, A2  A4,  A2  A5 ,  A3  A4,  A3  A5,  A4  A5.

Adding a sixth transversal gives us a total of fifteen parallelograms.

Figure 6: seven transversals

A1 , A2 , A3, A4, A5 , A6, A1  A2, A1  A3, A1  A4 , A1  A5, A1  A6 , A2  A3, A2  A4,  A2  A5 , A2  A6,  A3  A4,  A3  A5, A3  A6,  A4  A5, A4  A6, A5  A6.

Adding a seventh transversals gives us a total of twenty-one parallelograms.

Table 1: Side by Side view of Corresponding Transversals to Parallelograms

The general formula needs to be deduces from the patterns that are seen in the table and previously discovered and previously discovered maths formula. To discover the relationship between parallelograms and the number of vertical transversals a similar sequence of numbers needs to be investigated.

Graph  1

--> Let n = number of transversals and let p = number of parallelograms

Transversals (n)
Parallelograms (p)

n=2
n=3
n=4
n=5
n=6
n=7

n

p=1

p=3 (1 + 2)
p=6 (1 + 2 + 3)
p=10 (1 + 2 + 3 + 4)
p=15 (1 + 2 + 3 + 4 + 5)
p=21 (1 + 2 + 3 + 4 + 5 + 6)
1 + 2 + ... + (n - 1)

#### USE OF TECHNOLOGY

Using the TI - 84 Plus, press  STAT --> 1: Edit.

Type in L1, L2: (2, 1)

(3, 3)

(4, 6)

...etc.

Screen Capture from TI-84

Paralleograms:

1,        3,        6,        10,        15,        21,         etc.

3-1          6-3       10-6      15-15     21-15

2             3              4            5              6

1               1           ...