• 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

When parallel lines are intersected by parallel transversals to form parallelograms, relationships can be drawn between the number of parallelograms, transversals, and parallel lines. The number of parallelograms follow a triangular sequence when two parallel lines are intersected by transverals. In contrast, when three parallel lines are intersected by transversals, the number of parallelograms follow a tetrahedral number sequence.  

Figure 1 below shows a pair of horizontal parallel lines and four parallel transversals. Six parallelograms are formed: image15.png,image16.png,image22.png,image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png, image15.pngimage17.pngimage22.png.

image00.pngimage00.pngimage00.pngimage00.png

image14.pngimage03.pngimage13.pngimage05.pngimage04.png

Figure 1. Fourth parallel transversal crossing two parallel lines

Sum of Parallelogram(s)

Evidence

1

image15.png,image16.png,image22.png

2

image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png

3

image15.pngimage17.pngimage22.png

A fifth parallel transversal is added to the diagram as shown in Figure 2. 10 parallelograms are formed: image15.png,image16.png,image22.png,image18.png,image15.pngimage17.pngimage22.png,image16.pngimage17.pngimage18.png, image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png,image22.pngimage17.pngimage18.png,image15.pngimage17.pngimage18.png

image01.pngimage02.pngimage01.pngimage01.pngimage02.png

image04.pngimage05.pngimage03.pngimage07.pngimage07.pngimage06.png

Figure 2.

...read more.

Middle

image08.pngimage08.pngimage08.pngimage08.pngimage08.pngimage08.png

image09.png

image20.pngimage21.pngimage22.pngimage18.pngimage19.png

image10.png

Figure 3. Sixth transversal crossing two parallel lines

Sum of Parallelogram(s)

Evidence

1

image15.png,image16.png,image22.png,image18.png,image19.png

2

image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png,image22.pngimage17.pngimage18.png, image18.pngimage17.pngimage19.png.

3

image15.pngimage17.pngimage22.png,image16.pngimage17.pngimage18.png, image22.pngimage17.pngimage19.png

4

image15.pngimage17.pngimage18.png,image16.pngimage17.pngimage19.png

5

image15.pngimage17.pngimage19.png

A seventh parallel transversal is added to the diagram as shown in Figure 4. Fifteen parallelograms are formed:image15.png,image16.png,image22.png,image18.png,image19.png,image23.png,image15.pngimage17.pngimage23.png,image15.pngimage17.pngimage19.png,image16.pngimage17.pngimage23.png,image15.pngimage17.pngimage18.png, image16.pngimage17.pngimage19.png, image22.pngimage17.pngimage23.png,image15.pngimage17.pngimage22.png,image16.pngimage17.pngimage18.png, image22.pngimage17.pngimage19.png,image18.pngimage17.pngimage23.png,image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png,image22.pngimage17.pngimage18.png, image18.pngimage17.pngimage19.png,image19.pngimage17.pngimage23.png

image11.pngimage11.pngimage11.pngimage11.pngimage11.pngimage11.pngimage11.png

image12.png

image20.pngimage21.pngimage22.pngimage18.pngimage19.pngimage23.png

image12.png

Figure 4.

...read more.

Conclusion


image30.png = image31.png= t-th triangular number

image32.png= summation of a t-th triangular number(s) from 1 to n

image33.png=image34.png

image35.png=image36.png

= image37.png

=image38.pngimage39.pngimage40.png

=image38.pngimage41.pngimage42.png

=image43.png


image34.pngimage44.png=


image45.png=

image46.png


Thus this shows that if it's true for n, it's true for n + 1. Since we showed it

was true for n = 1, we now know it's also true for n = 1 + 1 = 2, and

then for n = 2 + 1 = 3, and so on, for all n >= 1.

Then, a general statement statement that satisfies both transversal and parallel lines can be drawn in that m represents horizontal parallel lines in n represents the intersected parallel transversals:

Suppose that:

image47.png=image48.png and image49.png=image50.png

Then:

image47.png* image49.png=image51.png

If m=2 in that there are 2 horizontal parallel lines and n=3 in that there are 3 parallel transversals

Then,

image47.png* image49.png= image52.png

=image53.png

=image54.png

=3

The conclusion that 3 parallelograms formed when 2 horizontal parallel lines are intersected by 3 parallel transversals is valid. Therefore, the general statement validity is true. The limitations of the equation image47.png* image49.png=image51.png is that it only considers the number of parallelograms formed by intersecting parallel lines. However, the equation gives an accurate result of when m represents horizontal parallel lines in n represents the intersected parallel transversals.

...read more.

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

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Maths SL Portfolio - Parallels and Parallelograms

    A8 ? A9 = 5 - A1 ? A2 ? A3 ? A4 ? A5 ? A6, - A2 ? A3 ? A4 ? A5 ? A6 ? A7, - A3 ? A4 ? A5 ? A6 ? A7 ? A8, - A4 ? A5 ? A6 ? A7 ? A8 ? A9 = 4 - A1 ?

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

    the general formula are the same as worked out earlier when not applying any formula at all; hence, it is correct and can be applied to terms 3 and 5. This was repeated using yet again another value for p, this time changing the p value from 5 to 4, resulting in the diagrams below.

  1. 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.

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

    We take the first three rows of the table and analyze them. 2. The pattern starts at 0 and shows that the # of parallelograms for certain # of transversal lines is the cumulative sum of the # of transversal lines before it.

  1. Parallels and Parallelograms

    Vertical Transversals Parallelograms formed 2 1 3 3 4 6 5 10 6 15 7 21 The similarity of these sequences is demonstrated in graph 1.1. Graph 1.1 The relationship between triangular numbers and the number of parallelograms formed when the number of vertical transversals is increased is clearly shown

  2. Series and Induction

    - n n2 + n = 2 x (?n) ?n = (n2 + n)/2 = n(n + 1)/2 We know that n3 - (n-1)3 = n3 - n3 + 3n2 - 3n + 1 = 3n2 - 3n + 1 Thus, 13 - 03 = 3(1)2 - 3(1)

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

    The graph : y = 1.22x 2 ? 6x + 11 The value of D for this graph is 6.56 The graph : y = 1.23x 2 ? 6x + 11 The value of D for this graph was 6.5 This process was repeated for many other graphs including y

  2. MATH IB SL INT ASS1 - Pascal's Triangle

    In order to find the explicit formula of the denominator I used the same strategy as for the numerator. First of all, I plotted the denominators of the 5th row up to the 9th in a table[2]. These numbers were calculated by using the recursive formula for Yn(r). r Y5(r)

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