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Parallels and Parallelograms

Extracts from this essay...

Introduction

QASMT Mathematics IA Parallels and Parallelograms Jeremiah Joseph 24/05/2009 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 1.1 Figure 1.2 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

Middle

? 3 = 3 Furthermore, The above examples justify that 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 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

Conclusion

Figure 6.1 The above figure has parallelograms A1, A2, A3, A4, A1 U A2, 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 6.2 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 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 ? 2 and ? 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 Furthermore, the variables, and 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, intersect with a constant number of horizontal lines to produce Av parallelograms. Furthermore, a varying number of horizontal lines, , intersect with a constant number of vertical transversals to produce Ah parallelograms. Therefore a varying number of vertical transversals, , intersect with a varying number of horizontal lines, , to produce Av × Ah parallelograms.

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