• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Parallels and Parallelograms

Extracts from this document...

Introduction

Jeremiah Joseph         Mr Peter Ellerton

 QASMT Mathematics IA Parallels and Parallelograms Jeremiah Joseph 24/05/2009 This internal assessment will investigate the relationship between vertical transversals, horizontal lines and parallelograms. Vertical transversals are lines that intersect horizontal lines. To create parallelograms two or more parallel vertical transversals needs to intersect with two or more horizontal lines. This is shown in figure 1.1 and figure 1.2.

Figure 1.1      Figure 1.2       These above figures demonstrate how vertical transversals (red) intersect with horizontal lines (black) to create parallelograms. These parallelograms are demonstrated in figure 1.1, A1, A2, A1 U A2. Furthermore, the parallelograms are illustrated in figure 1.2, A1, A2, A3, A1 U A2, A2 U A3, and A1 U A2 U A3. If parallel transversals are continually added an increasing number of parallelograms would be formed and a general formula can be deduced.

 Vertical Transversals Parallelograms Notation 2 1 A1 3 3 A1, A2, A1 U A2 4 6 A1, A2, A3,  A1 U A2, A2 U A3, A1 U A2 U A3 5 10 A1, A2, A3, A4, A1 U A2, A2 U A3, A3 U A4, A1 U A2 U A3 , A2 U A2 U A3, A1 U A2 U A3 U A4 6 15 A1, A2, A3, A4, A5, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A1 U A2 U A3 U A4 U A5 7 21 A1, A2, A3, A4, A5, A6, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A5 U A6, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A4 U A5 U A6, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5, A2 U A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5 U A6

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. 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.

Figure 2.1 The relationship between Tn and n is demonstrated in the picture above and the table below.

 Tn n 1 1 2 3 3 6 4 10 5 15 6 21

The relationship between the number of vertical transversals and the number of parallelograms formed is shown below.

 Vertical Transversals Parallelograms formed 2 1 3 3 4 6 5 10 6 15 7 21

Middle To combat this effect the equation needs to be changed to  , this is because a parallelogram is only created when there is two or more vertical transversal.

The general formula is:  .

Where v is the number of vertical transversal and P is the number of parallelograms formed.

However, this equation assumes that the number of horizontal lines stays constant at 1.

This formula is proved below.   3 = 3

Furthermore,  The above examples justify that  is the general formula for the relationship between the number of vertical transversals and the number of parallelograms. However, to develop a general formula that encompasses the variables of the number of vertical transversals and horizontal lines, an equation needs to be developed that deduces the number of parallelograms that is formed when horizontal lines vary. This equation needs to assume that the number of vertical transversals remain constant. The following figures demonstrate how additional parallelograms are formed when an increasing number of horizontal lines are added.

Figure 3.1                                                Figure 3.2          Figures 3.1 and 3.2 demonstrate how horizontal lines (black) are added to a constant number of vertical transversals to form an increasing number of parallelograms. This relationship is extrapolated in the table below.

 Horizontal lines Parallelograms Notation 2 1 A1 3 3 A1, A2, A1 U A2 4 6 A1, A2, A3,  A1 U A2, A2 U A3, A1 U A2 U A3 5 10 A1, A2, A3, A4, A1 U A2, A2 U A3, A3 U A4, A1 U A2 U A3 , A2 U A2 U A3, A1 U A2 U A3 U A4 6 15 A1, A2, A3, A4, A5, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A1 U A2 U A3 U A4 U A5 7 21 A1, A2, A3, A4, A5, A6, A1 U A2, A2 U A3, A3 U A4, A4 U A5, A5 U A6, A1 U A2 U A3, A2 U A3 U A4, A3 U A4 U A5, A4 U A5 U A6, A1 U A2 U A3 U A4, A2 U A3 U A4 U A5, A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5, A2 U A3 U A4 U A5 U A6, A1 U A2 U A3 U A4 U A5 U A6

Conclusion

2, A3 U A4, A1 U A3, A2 U A4 and A1 U A2 U A3 U A4. In total, figure 6.1 has 9 parallelograms; this coincides with the general formula’s answer.

Figure 6.2       Figure 6.2 has parallelograms A1, A2, A3, A1 U A2, A2 U A3, and A1 U A2 U A3. In total, figure 6.2 has 6 parallelograms. This coincides with the general formula’s answer.

Therefore the general statement for the relationship between vertical transversals, horizontal lines and the number of parallelograms is  Where P is the number of parallelograms produced, v is the number of vertical transversals and h is the number of horizontal lines.

However, this equation can only be used when  ≥ 2 and  ≥ 2. This is because a parallelogram is only created when 2 or more of each transversal are present. This is further demonstrated in figures 7.1 and 7.2.

Figure 7.1                                        Figure 7.2   Furthermore, the variables,  and  have to be natural numbers. The equation does not work when the variables are fractions, negative numbers or imaginary numbers.

The generalisation of the formula that was found was arrived at because a varying number of vertical transversals,  intersect with a constant number of horizontal lines to produce Av parallelograms. Furthermore, a varying number of horizontal lines,  , intersect with a constant number of vertical transversals to produce Ah parallelograms. Therefore a varying number of vertical transversals,  , intersect with a varying number of horizontal lines,  , to produce Av×Ah parallelograms.

Bibliography

Date Accessed: 22/05/09

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. ## Math Studies I.A

Reference http://www.president.gov.af/ http://uk.oneworld.net/guides/zimbabwe/development http://www.state.gov/r/pa/ei/bgn/2859.htm http://www.crisisgroup.org/home/index.cfm?id=1229 https://www.cia.gov/library/publications/the-world-factbook/geos/ni.html http://www.gov.sz/ Human Development Index (HDI) HDI uses GDP as a part of its calculation and then factors in indicators of life expectancy and education levels. http://en.wikipedia.org/wiki/Gross_domestic_product https://www.cia.gov/library/publications/the-world-factbook/fields/2004.html?countryCode=AF&rankAnchorRow=#AF (GDP? per capita (PPP)) http://unstats.un.org/unsd/demographic/products/indwm/tab3a.htm (life expectancy)

2. ## Math IA - Logan's Logo

The total height of the sine curve is 4.4 units, and divide this by 2 to get 2.2 (variable a). However, when we graph, it is clear that this line is not the center line of the curve: From here, we must then add the minimum y-value (-3.5)

1. ## Stellar Numbers. After establishing the general formula for the triangular numbers, stellar (star) shapes ...

method works out the same general formula, and this is tested below to assess the validity. Thereafter polysmlt on the graphic display calculator is used in order to retrieve the values of a, b, and c. The coefficients above are simply plugged into the calculator; this can be seen in

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

21 6 18 3 Table 1.1 Now the 'Second Order Difference' is 3 - triple the first set of parallelograms (pair of parallels intersecting with parallel transversals). Due to the second order being three, I deduced and found true that the number of parallelograms was increasing in multiples of three.

1. ## Math Studies - IA

In the Masters (one of four majors) their mean score was 72. In the same way, the entire team's mean score can be determined in the majors. It is unlikely that all Ryder Cup players participate consequently, and that will be a limitation to the investigation. Concerning the Ryder Cup, Europe has won every time on this side of the millennium.

2. ## MATH IA- Filling up the petrol tank ARWA and BAO

For Bao’s Vehicle: E2 =p2(r +d× day/w)/f l =p2(20km+d/10)/(20km) ∴p2 =(E2×20km)/(20km+d/10) y=p2 and x=d ∴The lines are of the form y=(E2×20)/(20+x/10), where E2 is a constant. ∴The line is an inverse function. We observe that as d is increasing, p2 is decreasing and hence p2 is indirectly proportional to d when E2 is constant.

1. ## Investigating Slopes Assessment

will be f1(x)=3X2 1. For this one, I will continue with my sequence of number. Therefore, I will still equal my ?n? number 3, however, my ?a? will now equal to 2. So, with letter ?a? equalling to 2 my function will now be f(x)=2X3 f(x)= 2X3 X=?

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

Only one line passes through the graph when the value of a increases above 1.12. After this only the line y=2x passes through the graph of y=ax2+bx+c. How the conjecture was found out is shown below. First the graph of y = 1.12x 2 ? 6x + 11 was drawn. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 