• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
17. 17
17

# 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

Middle

A4  A5  A6 A7 A8 and

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:

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Mathematics SL Parellels and Parallelograms. This task will consider the number of parallelograms formed ...

tried out the formula I found out that the answers didn't match: U2 = 3 (3 +1) = 6 2 U3 = 4 (4 +1) = 10 2 The results we had have moved one term so instead of adding 1 to "n", we need to subtract 1 to "n" so the values can match.

2. ## Stellar Numbers. In this task geometric shapes which lead to special numbers ...

In this example a= = 6 Now that I know that the first part of the formula is 6n2 I can proceed to find the values of 'b' and 'c' just as in the first step. pSn 6S0 6S1 6S2 6S3 6S4 6S5 6S6 Sequence 1 13 37 73 121

1. ## Parallels and Parallelograms. Aim: To find the effects of increasing ...

After doing this, we can say that: N0 of transversal lines N0 of parallelograms 1 0 2 1 3 3 4 6 5 10 6 15 7 21 By using the computer program Autograph we are able to illustrate the existing relation between # transversal lines (x)

2. ## Parallelograms. This investigation will focus on the number of parallelograms formed by intersecting lines ...

A5, A2 ? A3 ? A4 ? A5 ? A6 and A1 ? A2 ? A3 ? A4 ? A5 ? A6 The information above can be gathered in a table, as shown below. Transversals Parallelogram(s) 2 1 3 3 4 6 5 10 6 15 7 21 From this data we can generate a graph using excel, in

1. ## This essay will examine theoretical and experimental probability in relation to the Korean card ...

Nan-Cho, June is Mo- Ran, July is Hong-Ssa-Ri, August is Gong-San, September is Kook-Jun, October is Dan-Feng, November is Oh-Dong and December is Bi. This game "Sut-Da" does not use all these 48 cards but only uses 20 cards. This game can hold two to ten people in one game,

2. ## Parallels and Parallelograms

A similar sequence of numbers is present in Pascal's theory of triangular numbers. Therefore this sequence of numbers needs to be investigated to determine its similarity to the relationship between vertical transversals and the number of parallelograms formed. This sequence of numbers and the general formula of triangular numbers is shown in figure 2.1.

1. ## Parabola investigation. The property that was investigated was the relationship between the parabola and ...

4.52 = 1.63 X3 = 4.52 D = 0.99 X4 = 6.15 The graph : y = 1.02x 2 ? 6x + 11 X1 = 1.78 X2 = 2.44 SL = 0.66 SR = 1.64 X3 = 4.43 D = 0.98 X4 = 6.07 The value of ?a? The value

2. ## Parallels and Parallelograms Maths Investigation.

p=10 (1 + 2 + 3 + 4) p=15 (1 + 2 + 3 + 4 + 5) p=21 (1 + 2 + 3 + 4 + 5 + 6) 1 + 2 + ... + (n - 1) USE OF TECHNOLOGY Using the TI - 84 Plus, press STAT --> 1: Edit.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to