• 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

# 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: ,,,,, .

Figure 1. Fourth parallel transversal crossing two parallel lines

 Sum of Parallelogram(s) Evidence 1 ,, 2 , 3

A fifth parallel transversal is added to the diagram as shown in Figure 2. 10 parallelograms are formed: ,,,,,, ,,,

Figure 2.

Middle

Figure 3. Sixth transversal crossing two parallel lines

 Sum of Parallelogram(s) Evidence 1 ,,,, 2 ,,, . 3 ,, 4 , 5

A seventh parallel transversal is added to the diagram as shown in Figure 4. Fifteen parallelograms are formed:,,,,,,,,,, , ,,, ,,,,, ,

Figure 4.

Conclusion

= = t-th triangular number

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

=

=

=

=

=

=

=

=

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:

= and =

Then:

* =

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

Then,

* =

=

=

=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 * = 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.

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. ## 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 ...

To make my statements broader I expanded my table to the following (table 1.2): 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

1. ## 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.

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

Hence the formula for this sequence is y = .5x2 - .5x In order to determine the validity of this statement I have tested the formula with 2, 5, and 7 transversals. y = .5x2 - .5x y = .5(2)2 - .5(2)

1. ## Infinite Summation

Ä Uï¿½ ï¿½?ï¿½!ï¿½ï¿½"!ï¿½!ï¿½!ï¿½Xï¿½puB Lï¿½ 2ï¿½'ï¿½;ï¿½x ï¿½,PyWï¿½Bï¿½Rï¿½Dï¿½ï¿½L ï¿½y ~P^Qï¿½ï¿½@ï¿½ï¿½A1dï¿½-ï¿½Aï¿½(y!Lï¿½ï¿½ Mï¿½S "O>ï¿½[email protected]ï¿½L ï¿½< \ï¿½Bï¿½yLmvLï¿½LXTï¿½ Qï¿½ï¿½PK !ï¿½ï¿½fÈ¨ï¿½word/media/image52.pngï¿½PNG IHDR ï¿½M\ï¿½%iCCPICC Profilex..."MHaï¿½ï¿½ï¿½ï¿½ï¿½-ï¿½ï¿½\$T& Rï¿½+Sï¿½eï¿½L bï¿½}wï¿½gï¿½(tm)ï¿½-E""ï¿½uï¿½.VDï¿½ï¿½Nï¿½Cï¿½:D(tm)uï¿½ ï¿½E^"ï¿½ï¿½;""cT3/403ï¿½yï¿½ï¿½ï¿½|1/2ï¿½Uï¿½Rï¿½cE4`ï¿½ï¿½"ï¿½ÞvztLï¿½Uï¿½F\)ï¿½s:ï¿½ï¿½(c)ï¿½ï¿½kï¿½-iYj"ï¿½ï¿½6|"v(tm)P4*wd>,y<ï¿½ï¿½'/ï¿½<5g\$(c)4ï¿½!7ï¿½Cï¿½Nï¿½-ï¿½ï¿½lï¿½ï¿½Cï¿½ï¿½Tï¿½S"3-q";ï¿½-E#+c> ï¿½vÚ´ï¿½ï¿½=ï¿½SÔ°ï¿½ï¿½79 Ú¸ï¿½@ï¿½-`Óï¿½mï¿½-ï¿½vï¿½Ulï¿½5ï¿½ï¿½`ï¿½Pï¿½=ï¿½ï¿½Gï¿½ï¿½ï¿½jï¿½ï¿½)ï¿½kï¿½P*}ï¿½6ï¿½~^/ï¿½~ï¿½.ï¿½~ï¿½a ï¿½ï¿½ï¿½2 nï¿½×²0ï¿½%ï¿½ï¿½fï¿½ï¿½ï¿½ï¿½ï¿½ï¿½|U ï¿½ï¿½9ï¿½lï¿½ï¿½7?ï¿½ï¿½ï¿½j`ï¿½'ï¿½l7ï¿½ï¿½ï¿½"ï¿½tï¿½iï¿½ï¿½Nï¿½f]?ï¿½uï¿½h...ï¿½gM ZÊ²4ï¿½ï¿½i(r)ï¿½"[ï¿½&LYï¿½ï¿½_ï¿½xï¿½ {xï¿½Oï¿½ï¿½\$1/4ï¿½ß¬Ì¥S]ï¿½%ï¿½ï¿½Ö§ï¿½ï¿½ï¿½&7ï¿½ï¿½gÌ>r=ï¿½ï¿½*g8`ï¿½(tm)ï¿½ 8rÊ¶ï¿½<(c)ï¿½ï¿½ï¿½ï¿½ï¿½"dï¿½WT'"ï¿½<ï¿½ eLï¿½~.u"A(r)ï¿½=9(tm)ï¿½-ï¿½]ï¿½ï¿½>31ï¿½3'ï¿½X3ï¿½ï¿½ï¿½ï¿½-\$eï¿½}ï¿½ï¿½u,ï¿½ï¿½gm'g...6ï¿½64\$Ñï¿½E zL*LZï¿½_ï¿½jï¿½ï¿½ï¿½_ï¿½ï¿½]1/2Xï¿½ï¿½yï¿½[ï¿½?...Xs ï¿½ï¿½ï¿½Nï¿½ï¿½/ï¿½ ï¿½]ï¿½ï¿½|mï¿½3/4ï¿½(tm)sÏï¿½"k_Wf-ï¿½È¸Aï¿½23/4ï¿½)ï¿½oï¿½ï¿½ z-diï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½2ï¿½|mÙ£ï¿½ï¿½j|5Ô¥ejï¿½8ï¿½ï¿½(r)eï¿½Eï¿½ï¿½7ï¿½ï¿½[ï¿½ï¿½ï¿½Qï¿½|ï¿½IM%×²ï¿½xf)ï¿½|6\ kï¿½ï¿½"`Ò²"ï¿½ä.<kï¿½ï¿½Uï¿½}jï¿½ï¿½ M=ï¿½ï¿½"mjßï¿½ï¿½Üï¿½ï¿½ ï¿½ï¿½eï¿½)ï¿½`cï¿½ï¿½ï¿½IWfï¿½ï¿½ï¿½ï¿½/ï¿½ï¿½ï¿½^a ï¿½44ï¿½Mï¿½ï¿½ï¿½i ï¿½ï¿½6pï¿½"ï¿½ï¿½_ï¿½ Þ¡ï¿½ï¿½>IDAT8ï¿½';KQ...ï¿½ï¿½ï¿½)Dï¿½ï¿½ï¿½`'ï¿½`)Vbgkï¿½ï¿½>ï¿½ï¿½ ï¿½tVZ[k' ï¿½ï¿½UDWKA"Xï¿½Yï¿½,ï¿½ï¿½aï¿½3ï¿½3/4ï¿½ï¿½ ï¿½o5ï¿½v~ï¿½)oï¿½"-Xï¿½ï¿½ï¿½Sï¿½ï¿½ï¿½p w&ï¿½ï¿½>Yï­°f_ï¿½}X...3ï¿½Aï¿½ï¿½@ï¿½?Eu^ï¿½ <ï¿½i"d -ï¿½[email protected]ï¿½ mf R ï¿½ï¿½Iï¿½ï¿½.}R6`ï¿½!...9ï¿½ï¿½ï¿½ Éºï¿½,ï¿½&Û£Xï¿½lï¿½uï¿½ï¿½ï¿½x%?ï¿½Tï¿½ï¿½ï¿½ï¿½`ï¿½Ð6ï¿½ï¿½Wï¿½Teï¿½ï¿½ ï¿½Jï¿½ï¿½Sï¿½(r)ï¿½lï¿½Z2"ï¿½ï¿½ï¿½ ï¿½ï¿½ï¿½ï¿½86ï¿½ï¿½×£ï¿½ï¿½Lï¿½ï¿½]ï¿½ ï¿½ï¿½Rï¿½Aï¿½Bï¿½&q^ï¿½ï¿½[email protected] ï¿½ï¿½ ...ï¿½-ï¿½"ï¿½(tm)ï¿½ï¿½Aï¿½5Þ¤ 5ï¿½ ï¿½Eï¿½ï¿½ï¿½P(r)

2. ## Parallels and Parallelograms

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.

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

For each case there is a separate conjecture. The following is the conjecture for the first case that is ?Both the lines intersect? The conjecture is written below. ?As ?a? is increased by 0.01, D reduces by 0.01?. How this conjecture was found out is shown below.

2. ## Parallels and Parallelograms Maths Investigation.

Graph 1 --> Let n = number of transversals and let p = number of parallelograms Transversals (n) Parallelograms (p) n=2 n=3 n=4 n=5 n=6 n=7 n p=1 p=3 (1 + 2) p=6 (1 + 2 + 3) p=10 (1 + 2 + 3 + 4)

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