- Level: International Baccalaureate
- Subject: Maths
- Word count: 1062
Parallels and Parallelograms
Extracts from this document...
Introduction
When parallel lines are intersected by parallel transversals to form parallelograms, relationships can be drawn between the number of parallelograms, transversals, and parallel lines. The number of parallelograms follow a triangular sequence when two parallel lines are intersected by transverals. In contrast, when three parallel lines are intersected by transversals, the number of parallelograms follow a tetrahedral number sequence.
Figure 1 below shows a pair of horizontal parallel lines and four parallel transversals. Six parallelograms are formed: ,,,,, .
Figure 1. Fourth parallel transversal crossing two parallel lines
Sum of Parallelogram(s) | Evidence |
1 | ,, |
2 | , |
3 |
A fifth parallel transversal is added to the diagram as shown in Figure 2. 10 parallelograms are formed: ,,,,,, ,,,
Figure 2.
Middle
Figure 3. Sixth transversal crossing two parallel lines
Sum of Parallelogram(s) | Evidence |
1 | ,,,, |
2 | ,,, . |
3 | ,, |
4 | , |
5 |
A seventh parallel transversal is added to the diagram as shown in Figure 4. Fifteen parallelograms are formed:,,,,,,,,,, , ,,, ,,,,, ,
Figure 4.
Conclusion
= = t-th triangular number
= summation of a t-th triangular number(s) from 1 to n
=
=
=
=
=
=
=
=
Thus this shows that if it's true for n, it's true for n + 1. Since we showed it
was true for n = 1, we now know it's also true for n = 1 + 1 = 2, and
then for n = 2 + 1 = 3, and so on, for all n >= 1.
Then, a general statement statement that satisfies both transversal and parallel lines can be drawn in that m represents horizontal parallel lines in n represents the intersected parallel transversals:
Suppose that:
= and =
Then:
* =
If m=2 in that there are 2 horizontal parallel lines and n=3 in that there are 3 parallel transversals
Then,
* =
=
=
=3
The conclusion that 3 parallelograms formed when 2 horizontal parallel lines are intersected by 3 parallel transversals is valid. Therefore, the general statement validity is true. The limitations of the equation * = is that it only considers the number of parallelograms formed by intersecting parallel lines. However, the equation gives an accurate result of when m represents horizontal parallel lines in n represents the intersected parallel transversals.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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