- Level: International Baccalaureate
- Subject: Maths
- Word count: 3688
Parallels and Parallelograms Maths Investigation.
Extracts from this document...
Introduction
CAYMAN INTERNATIONAL SCHOOL |
PARALLELS AND PARALLELOGRAMS |
JULIANA N. WOOD |
9/7/2012 |
Definitions:
Transversal: A 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: A 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 A1ᴗA2.
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,A1ᴗA2,A2ᴗA3 and A1ᴗA3.
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, A1ᴗA2, A1ᴗA3, A1ᴗA4, A1ᴗA5, A1ᴗA6, A2ᴗA3, A2ᴗA4, A2ᴗA5, A2ᴗA7, A3ᴗA4, A3ᴗA5, A3ᴗA8, A4ᴗA5, A4ᴗA9, A5ᴗA10, A6ᴗA7, A6ᴗA8, A6ᴗA9, A6ᴗA10, A7ᴗA8, A7ᴗA9, A7ᴗA10, A8ᴗA9, A8ᴗA10, A9ᴗA10, A1ᴗA7, A1ᴗA8, A1ᴗA9, A2ᴗA8, A2ᴗA9 ,A2ᴗA10, A3ᴗA9, A3ᴗA10, A4ᴗA10, A1ᴗA10.
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, A1ᴗA2, A1ᴗA3, A1ᴗA4, A1ᴗA5, A1ᴗA6 , A1ᴗA7 A2ᴗA3, A2ᴗA4, A2ᴗA5, A2ᴗA6, A2ᴗA8, A3ᴗA4, A3ᴗA5, A3ᴗA6, A3ᴗA9, A4ᴗA5, A4ᴗA6, A4ᴗA10, A5ᴗA6, A5ᴗA11, A6ᴗA12, A7ᴗA8, A7ᴗA9, A7ᴗA10, A7ᴗA11, A7ᴗA12, A8ᴗA9, A8ᴗA10, A8ᴗA11, A8ᴗA12, A9ᴗA10, A9ᴗA11, A9ᴗ
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
2 horizontal lines:1nn-1÷2
3 horizontal lines:3(nn-1)÷2
4 horizontal lines:6(nn-1)÷2
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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