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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 image21.pngimage21.png , (image87.pngimage87.png) 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.

image00.pngimage00.png

image02.png

            A1

image02.png

        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.

image01.pngimage01.pngimage00.pngimage00.pngimage00.png

        A1        A2

        Figure 2

Task 1:

...read more.

Middle

 A4  A5  A6 A7 A8 and

A1 A2  A3  A4  A5 A6  A7 A8.

image00.pngimage00.pngimage00.pngimage00.pngimage00.pngimage00.pngimage00.pngimage00.pngimage00.png

image11.png

                  A1        A2         A3         A4        A5          A6        A7         A8

image12.png

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: image122.pngimage122.png

Number of transversals n = 11

Therefore: image126.pngimage126.png

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

...read more.

Conclusion

Task 4: Now extend your results to m horizontal parallel lines intersected by n parallel transversals.

The number of parallelograms visible, when image33.pngimage33.png horizontals and image34.pngimage34.png transversals are intersected, image35.pngimage35.png 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 image36.pngimage36.png.

My Example Figure 20 examines the case that there are five vertical and four horizontal lines:

Subpart size (in fields)

Number of possible parallelograms

image37.png

width (image38.pngimage38.png)

height (image39.pngimage39.png)

image40.png

image40.png

image42.png

image43.png

image40.png

image44.png

image45.png

image46.png

image40.png

image47.png

image48.png

image49.png

image44.png

image40.png

image50.png

image51.png

image44.png

image44.png

image52.png

image53.png

image44.png

image47.png

image54.png

image55.png

image47.png

image40.png

image56.png

image57.png

image47.png

image44.png

image58.png

image59.png

image47.png

image47.png

image60.png

image61.png

image62.png

image40.png

image63.png

image64.png

image62.png

image44.png

image66.png

image67.png

image62.png

image47.png

image68.png

image69.png

                                                                                           Figure 20

The example shows that the number of parallelograms (image70.pngimage70.png) of a certain size image71.pngimage71.png can be determined using the following rule:

image72.png

If one adds all image73.pngimage73.pngimage35.pngimage35.png is obtained:

image74.png
image75.png
image76.png

image77.png

image79.png
image80.png

image81.png
image82.png

image83.pngimage83.png
image84.png

To verify my general statement I’ll show how it works by using two examples.

Example 3:

General statement: image85.pngimage85.png

Number of transversals: image86.pngimage86.png

Number of horizontal lines: himage88.pngimage88.png

image89.png

image90.png

image91.png

image92.png

image93.png

Figure 17 also verifies that my statement works.

Example 4:

General statement: image85.pngimage85.png

Number of transversals: image86.pngimage86.png

Number of horizontal lines: himage95.pngimage95.png2

image96.png

image97.png

image98.png

image99.png

image100.png

image101.png

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:

  1. We cannot have negative numbers.
  2. There are no negative transversals or horizontals .
  3. There are no transversals or horizontals with a fraction.

image102.png

image103.png

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]!!!

...read more.

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

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