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Parallels and Parallelograms

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Introduction

Jeremiah Joseph         Mr Peter Ellerton

QASMT

Mathematics IA

Parallels and Parallelograms

Jeremiah Joseph

24/05/2009

image24.png


This internal assessment will investigate the relationship between vertical transversals, horizontal lines and parallelograms. Vertical transversals are lines that intersect horizontal lines. To create parallelograms two or more parallel vertical transversals needs to intersect with two or more horizontal lines. This is shown in figure 1.1 and figure 1.2.

Figure 1.1image02.pngimage02.pngimage00.pngimage01.png

image04.pngimage09.png

Figure 1.2

image23.pngimage02.pngimage02.pngimage02.png

image03.pngimage04.png

image05.png

These above figures demonstrate how vertical transversals (red) intersect with horizontal lines (black) to create parallelograms. These parallelograms are demonstrated in figure 1.1, A1, A2, A1 U A2. Furthermore, the parallelograms are illustrated in figure 1.2, A1, A2, A3, A1 U A2, A2 U A3, and A1 U A2 U A3. If parallel transversals are continually added an increasing number of parallelograms would be formed and a general formula can be deduced.

Vertical Transversals

Parallelograms

Notation

2

1

A1

3

3

A1, A2, A1 U A2

4

6

A1, A2, A3,  A1 U A2, A2 U A3, A1 U A2 U A3

5

10

A1, A2, A3, A4, A1 U A2, A2 U A3, A3 U A4, A1 U A2 U A3

, A2 U A2 U A3, A1 U A2 U A3 U A4

6

15

A1, A2, A3, A4, A5, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A1 U A2 U A3 U A4 U A5  

7

21

A1, A2, A3, A4, A5, A6, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A5 U A6, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A4 U A5 U A6, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5, A2 U A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5 U A6

The general formula needs to be deduced from the patterns that are seen in the table and previously discovered maths formula. To discover the relationship between parallelograms and the number of vertical transversals a similar sequence of numbers needs to investigated. A similar sequence of numbers is present in Pascal’s theory of triangular numbers. Therefore this sequence of numbers needs to be investigated to determine its similarity to the relationship between vertical transversals and the number of parallelograms formed. This sequence of numbers and the general formula of triangular numbers is shown in figure 2.1.

Figure 2.1

image25.png

The relationship between Tn and n is demonstrated in the picture above and the table below.

Tn

n

1

1

2

3

3

6

4

10

5

15

6

21

The relationship between the number of vertical transversals and the number of parallelograms formed is shown below.

Vertical Transversals

Parallelograms formed

2

1

3

3

4

6

5

10

6

15

7

21

...read more.

Middle

image07.png

To combat this effect the equation needs to be changed toimage44.pngimage44.png, this is because a parallelogram is only created when there is two or more vertical transversal.

The general formula is:image26.pngimage26.png.

Where v is the number of vertical transversal and P is the number of parallelograms formed.

However, this equation assumes that the number of horizontal lines stays constant at 1.

This formula is proved below.

image27.png

image28.pngimage28.png

3 = 3

Furthermore,

image29.png

image30.png

The above examples justify that image31.pngimage31.png is the general formula for the relationship between the number of vertical transversals and the number of parallelograms. However, to develop a general formula that encompasses the variables of the number of vertical transversals and horizontal lines, an equation needs to be developed that deduces the number of parallelograms that is formed when horizontal lines vary. This equation needs to assume that the number of vertical transversals remain constant. The following figures demonstrate how additional parallelograms are formed when an increasing number of horizontal lines are added.

Figure 3.1                                                Figure 3.2

image09.pngimage08.png

image10.png

image09.pngimage04.pngimage08.pngimage11.pngimage04.pngimage12.pngimage08.png


Figures 3.1 and 3.2 demonstrate how horizontal lines (black) are added to a constant number of vertical transversals to form an increasing number of parallelograms. This relationship is extrapolated in the table below.

Horizontal lines

Parallelograms

Notation

2

1

A1

3

3

A1, A2, A1 U A2

4

6

A1, A2, A3,  A1 U A2, A2 U A3, A1 U A2 U A3

5

10

A1, A2, A3, A4, A1 U A2, A2 U A3, A3 U A4, A1 U A2 U A3 , A2 U A2 U A3, A1 U A2 U A3 U A4

6

15

A1, A2, A3, A4, A5, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A1 U A2 U A3 U A4 U A5  

7

21

A1, A2, A3, A4, A5, A6, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A5 U A6, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A4 U A5 U A6, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5, A2 U A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5 U A6

...read more.

Conclusion

2, A3 U A4, A1 U A3, A2 U A4 and A1 U A2 U A3 U A4. In total, figure 6.1 has 9 parallelograms; this coincides with the general formula’s answer.

Figure 6.2image20.pngimage20.png

image04.png

image21.png

image09.png

image21.png

image21.png

Figure 6.2 has parallelograms A1, A2, A3, A1 U A2, A2 U A3, and A1 U A2 U A3. In total, figure 6.2 has 6 parallelograms. This coincides with the general formula’s answer.

Therefore the general statement for the relationship between vertical transversals, horizontal lines and the number of parallelograms is image37.pngimage37.png

Where P is the number of parallelograms produced, v is the number of vertical transversals and h is the number of horizontal lines.

However, this equation can only be used when image38.pngimage38.png ≥ 2 and image40.pngimage40.png ≥ 2. This is because a parallelogram is only created when 2 or more of each transversal are present. This is further demonstrated in figures 7.1 and 7.2.

Figure 7.1                                        Figure 7.2

image06.pngimage02.png

image22.png

Furthermore, the variables, image41.pngimage41.pngand image40.pngimage40.png have to be natural numbers. The equation does not work when the variables are fractions, negative numbers or imaginary numbers.

The generalisation of the formula that was found was arrived at because a varying number of vertical transversals, image38.pngimage38.png intersect with a constant number of horizontal lines to produce Av parallelograms. Furthermore, a varying number of horizontal lines, image40.pngimage40.png, intersect with a constant number of vertical transversals to produce Ah parallelograms. Therefore a varying number of vertical transversals, image38.pngimage38.png, intersect with a varying number of horizontal lines, image40.pngimage40.png, to produce Av×Ah parallelograms.

Bibliography

http://upload.wikimedia.org/wikipedia/commons/3/33/Números_triangulares.png

Date Accessed: 22/05/09

...read more.

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