• 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 Maths Investigation.

Extracts from this document...

Introduction

 CAYMAN INTERNATIONAL SCHOOL PARALLELS AND PARALLELOGRAMS JULIANA N. WOOD
 9/7/2012

## Definitions:

Transversal: 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: 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 A1A2.

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,A1A2,A2A3  and A1A3.

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, A1A2, A1A3, A1A4, A1A5, A1A6, A2A3, A2A4, A2A5,  A2A7, A3A4, A3A5, A3A8, A4A5, A4A9, A5A10, A6A7, A6A8, A6A9, A6A10, A7A8, A7A9, A7A10, A8A9, A8A10, A9A10, A1A7, A1A8, A1A9, A2A8, A2A9 ,A2A10, A3A9, A3A10, A4A10, A1A10.

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, A1A2, A1A3, A1A4, A1A5, A1A6 , A1A7 A2A3, A2A4, A2A5, A2A6, A2A8, A3A4, A3A5, A3A6, A3A9, A4A5, A4A6, A4A10, A5A6, A5A11, A6A12, A7A8, A7A9, A7A10, A7A11, A7A12, A8A9, A8A10,  A8A11,  A8A12, A9A10,  A9A11,  A9

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

## 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.

## 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 ...

Now our final formula is: Un = n (n -1) 2 Once again I tested the above: U2 = 2 (2 -1)

2. ## Math Investigation - Properties of Quartics

Equating the equation of line and the quartic function with each other will give us all the four points of inflection P, Q, R, and S respectively. By using synthetic division and known roots we can find the equation of the quartic into quadratic as done previously, which can be solved using quadratic formula.

1. ## High Jump Gold Medal 2012 maths investigation.

y= 224.8 cm The answer are mathematical correct. However, in reality we do not know whether the data will truly describe the result since we only the data up to 1980. Data table from 1896 - 2008: Numbers of year Actual height/cm 0 190 8 180 12 191 16 193

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

U B U C U D, B U C U D U E, C U D U E U F, A U B U C U D U E, B U C U D U E U F, A U B U C U D U E U F.

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

A4, A3 ? A4 ? A5, A4 ? A5 ? A6, A1 ? A2 ? A3 ? A4, A2 ? A3 ? A4 ? A5, A3 ? A4 ? A5 ? A6, A1 ? A2 ? A3 ? A4 ? A5, A2 ? A3 ? A4 ? A5 ?

2. ## Parallels and Parallelograms

The general formula needs to be deduced from the patterns that are seen in the table and previously discovered maths formula. To discover the relationship between parallelograms and the number of vertical transversals a similar sequence of numbers needs to investigated.

1. ## Music and Maths Investigation. Sine waves and harmony on the piano.

The reason for picking one half is purely ease and simplicity, any other number less than one and greater than zero could have been used however, it is easiest to visualize a pattern 12. This proof variation was created by French mathematician, Nicole Oresme.

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

The three conditions are written below. 1. Both the two lines ?y=x? and ?y=2x? intersect the graph of y = ax2 + bx + c 2. Only one line that is ?y=2x? 3. None of the lines intersects(in this case, the value of D is 0)

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