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Parallels and Parallelograms Maths Investigation.

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Introduction

 CAYMAN INTERNATIONAL SCHOOL PARALLELS AND PARALLELOGRAMS JULIANA N. WOOD
 9/7/2012

Definitions:

Transversal: line that cuts across a set of lines or the sides of a plane figure. Transversals often cut across parallel lines.

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

Parallelogram: quadrilateral with two pairs of parallelsides.

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

Adding a third transversal

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

A2

A1

Figure 2: three transversals

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

Adding a fourth transversal Figure 3: four transversals

A1 , A2 , A3,A1A2,A2A3  and A1A3.

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

Middle

.

Adding a third horizontal parallel line with five transversal lines gives us a total of thirty parallelograms.

Adding a third horizontal parallel line with six transversals Figure 12: six transversals

A1 , A2, A3, A4, A5, A6, A7, A8, A9, A10, A1A2, A1A3, A1A4, A1A5, A1A6, A2A3, A2A4, A2A5,  A2A7, A3A4, A3A5, A3A8, A4A5, A4A9, A5A10, A6A7, A6A8, A6A9, A6A10, A7A8, A7A9, A7A10, A8A9, A8A10, A9A10, A1A7, A1A8, A1A9, A2A8, A2A9 ,A2A10, A3A9, A3A10, A4A10, A1A10.

Adding a third horizontal parallel line with six transversal lines gives us a total of forty-five parallelograms.

Adding a third horizontal parallel line with seven transversals Figure 13: seven transversals

A1 , A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A1A2, A1A3, A1A4, A1A5, A1A6 , A1A7 A2A3, A2A4, A2A5, A2A6, A2A8, A3A4, A3A5, A3A6, A3A9, A4A5, A4A6, A4A10, A5A6, A5A11, A6A12, A7A8, A7A9, A7A10, A7A11, A7A12, A8A9, A8A10,  A8A11,  A8A12, A9A10,  A9A11,  A9

Conclusion

General statement

Since the number of parallelograms created as the number of transversals increased each had a Second Order difference of 6, it was immediately known that the general formula must be a quadratic equation.

If there are n transversals and four horizontal lines, then p = sum of all integers from 1 to (n - 1).

P=12 4nn-1
or
P=4n2-n÷2

Expanding…

 Number of Parallel Transversals 2 3 4 5 6 7 n 2 1 3 6 10 15 21 3 3 9 18 30 45 63 4 6 18 36 60 90 126 5 10 30 60 100 150 210 6 15 45 90 150 225 345 7 21 63 126 210 315 441 m

Table 7: Number of Parallel Transversals

5 horizontal lines:10(nn-1)÷2

The number and sequence repeats the formula nn-1÷2 and multiplies by the first term.

The General Statement

From this realization I was able to find the final formula for calculating the number of parallelograms formed when m horizontal parallel lines are intersected by n parallel transversal:

m(m-1)2  ×  n(n-1)2

To test the validity of the formula, I tested it against a previously counted parallelograms (Figure 10), the intersection of four transversals with three horizontal parallel lines should form a total of eighteen parallelograms. Figure 14: four transversals

Using the formula

m(m-1)2  ×  n(n-1)2

3(3-1)2  ×  4(4-1)2

3(2)2  ×  4(3)2

62  ×  122

3 ×6

=18 Parallelograms

Scope/limitations

The formula will be valid for m, n≥2.if either value were to be 1 or 0, it would be impossible to create any parallelograms.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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