- Level: International Baccalaureate
- Subject: Maths
- Word count: 2433
Parallels and Parallelograms
Extracts from this document...
Introduction
Mathematics Standard Level
Portfolio by Emanuel Hausmann
Content
Task 1: Show that six parallelograms are formed when a fourth transversal is added to Figure 2. List all parallelograms, using set notation.
Task 2: Repeat the process with 5, 6 and 7 transversals. Show your results in a table. Use technology to find a relation between the number of transversals and the number of parallelograms. Develop a general statement, and test its validity.
Task 3: Next consider the number of parallelograms formed by three horizontal parallel lines intersected by parallel transversals. Develop and test another general statement for this case.
Task 4: Now extend your results to m horizontal parallel lines intersected by n parallel transversals.
Scope and Limitations:
Parallels and Parallelograms
In this portfolio task I considered the number of parallelograms formed by intersecting parallel lines. The general statement for the overall pattern is , () where C is the count, h the number of horizontals and t the number of transversals. In the following I’ll explain how I arrived at this generalization.
Figure 1 below shows a pair of horizontal parallel lines and a pair of parallel transversals. One parallelogram (A1) is formed.
A1
Figure 1
A third parallel transversal is added to the diagram as shown in Figure 2. Three parallelograms are formed: A1, A2 and A1∪ A2.
A1 A2
Figure 2
Task 1:
Middle
A1∪ A2 ∪ A3 ∪ A4 ∪ A5∪ A6 ∪ A7∪ A8.
A1 A2 A3 A4 A5 A6 A7 A8
Figure 11 ^{[a]}
To verify the general statement Figure 12 illustrates how many parallelograms are formed if we have 11 transversals and a pair of horizontal lines.
General term:
Number of transversals n = 11
Therefore:
The so formed parallelograms are: A1, A2, A3, A4, A5, A6, A7, A8, A9, A10,
A1∪ A2, A2∪ A3, A3∪ A4, A4∪ A5, A5∪ A6, A6∪ A7, A7∪ A8, A8∪ A9, A9∪ A10,
A1∪ A2 ∪ A3, A2∪ A3 ∪ A4, A3∪ A4 ∪ A5, A4∪ A5 ∪ A6 , A5∪ A6 ∪ A7 , A6∪ A7 ∪ A8 , A7∪ A8 ∪ A9, A8∪ A9 ∪ A10,
A1∪ A2 ∪ A3 ∪ A4, A2∪ A3 ∪ A4 ∪ A5, A3∪ A4 ∪ A5 ∪ A6 , A4∪ A5 ∪ A6 ∪ A7 , A5∪ A6 ∪ A7 ∪ A8 , A6∪ A7 ∪ A8 ∪ A9 , A7∪ A8 ∪ A9 ∪ A10 ,
A1∪ A2 ∪ A3 ∪ A4 ∪ A5, A2∪ A3 ∪ A4 ∪ A5 ∪ A6, A3∪ A4 ∪ A5 ∪ A6 ∪ A7, A4∪ A5 ∪ A6 ∪ A7 ∪ A8, A5∪ A6 ∪ A7 ∪ A8 ∪ A9, A6∪ A7 ∪ A8 ∪ A9 ∪ A10,
A1∪ A2 ∪ A3 ∪ A4 ∪ A5∪ A6, A2∪ A3 ∪ A4 ∪ A5 ∪ A6∪ A7, A3∪ A4 ∪ A5 ∪ A6 ∪ A7∪ A8, A4∪ A5 ∪ A6 ∪ A7 ∪ A8∪ A9, A5∪ A6 ∪ A7 ∪ A8 ∪ A9∪ A10,
A1∪ A2 ∪ A3 ∪ A4 ∪ A5∪ A6 ∪ A7, A2∪ A3 ∪ A4 ∪ A5 ∪ A6∪ A7∪ A8, A3∪ A4 ∪ A5 ∪ A6 ∪ A7∪ A8∪ A9
Conclusion
Task 4: Now extend your results to m horizontal parallel lines intersected by n parallel transversals.
The number of parallelograms visible, when horizontals and transversals are intersected, can be determined considering the following:
The overall sum of parallelograms is equal to the sum of the number of parallelograms for of all possible sizes. The maximum size of a parallelogram is .
My Example Figure 20 examines the case that there are five vertical and four horizontal lines:
Subpart size (in fields) | Number of possible parallelograms | ||
width () | height () | ||
Figure 20
The example shows that the number of parallelograms () of a certain size can be determined using the following rule:
If one adds all is obtained:
To verify my general statement I’ll show how it works by using two examples.
Example 3:
General statement:
Number of transversals:
Number of horizontal lines: h
Figure 17 also verifies that my statement works.
Example 4:
General statement:
Number of transversals:
Number of horizontal lines: h2
Figure 4 underlines the correctness of my general statement.
Scope and Limitations:
By looking at all my work I made some observations in order to find out the scope and limitations:
- We cannot have negative numbers.
- There are no negative transversals or horizontals .
- There are no transversals or horizontals with a fraction.
Figure 21
Figure 21 shows the graph I have plotted in order to visualize the general statement and its limitations.
The following sheet (Figure 22) shows a part of the the table I have used to plot the graph.
[a]!!!
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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