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

Beyond Pythagoras

Extracts from this document...

Introduction

Beyond Pythagoras

Introduction:

 During this investigation I will be trying to find out patterns and formulas relating to pythagorus’ theorem. One pattern he found with triangles was that the smallest and middle length sides squared added together to make the largest side squared. For example:

  3   + 4   = 5

because  3   = 3 x 3 = 9

  4   = 4 x 4 = 16

  5   = 5 x 5 = 25

so   3   + 4   = 9 + 16 = 25 = 5

 (smallest number)   + (middle number)   = (largest number)

 Another name for this is Pythagorean triples: a   + b   = c  

 I will continue with this investigation to find as many rules and formulas as possible, to see if this is a one off, or if it only occurs in certain triangles. Further into my investigation I will also look at triangles that don’t

...read more.

Middle

 The other 2 examples of triangles that fit the pattern, 5,  12,  13 and 7,  24,  25 can also be the lengths of right angled triangles:

 These are the results of the perimeter and area of all 3 triangles. Another 2 triangles have been added to the sequence in order to find a pattern and rule:

Code Name

n

1

2

3

4

5

 To complete this table I used the first 3 in the table to get a basic rules:

a = n + 2

b = n   + (n + 1)   - 1

c = n   + (n + 1)  

P = 4n   + 6n + 2

A = 2n   + 3n   + n

 I worked all the above formulas out using this method:

e.g. b

      n 1 2 3 4 5      

 4 12 24 40 60            <------- b

       8         12         16        20

  4 4 4                          <------- meant n   was involved

 This meant that n   was included in the formula. So I squared n and compared them:

...read more.

Conclusion

 Also, the only strand I could find a relevant formula for was a. Therefore I decided not to continue with this part of my investigation and move on to the next part.

 Next I looked into the Pythagorean triples that didn’t fit the rules and formulas I have already found, some of them have been included in previous tables, but I used them in the table below because they fit in.

Code Name

n

1

2

3

4

5

6

7

 These are the formulas u found and used to complete this table:

a = 3n

b = 4n

c = 5n

P = 12n

A = 6n

 I worked the formulas out using the same method as before.

...read more.

This student written piece of work is one of many that can be found in our GCSE Pythagorean Triples 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 GCSE Pythagorean Triples essays

  1. Maths GCSE coursework: Beyond Pythagoras

    Firstly we shall look at the perimeter: 12 30 56 90 132 182 18 26 34 42 50 1st difference 8 8 8 8 2nd difference The 2nd difference is 8 so we must find the formula for the 'perimeter'.

  2. Maths Number Patterns Investigation

    Middle Side = 220, Largest Side =221. I now have 10 different triangles, which I think is easily enough to find a relationship between each side. Shortest side2 = Middle side + Longest side The way I mentioned above and on the previous pages, is quite a good way of finding the middle and longest sides.

  1. Pythagoras Theorem

    = ([a� + 4]/4) ([a� + 4]/4) o = a� + (a� - 4)(a� - 4)/16 = (a� + 4)(a� + 4)/16 Cancel out the brackets: o (a� - 4)(a� - 4) = a4 - 4a� - 4a� + 16 = a4 - 8a� + 16 o (a� + 4)(a� + 4)

  2. Beyond Pythagoras

    87 3784 3785 44 89 3960 3961 45 91 4140 4141 46 93 4324 4325 47 95 4512 4513 48 97 4704 4705 49 99 4900 4901 50 101 5100 5101 51 103 5304 530 52 105 5512 5513 53 107 5724 5725 54 109 5940 5941 55 111 6160

  1. Beyond Pythagoras.

    I shall now try adding n to the formula to make the equation 2n? + n. 2 x 1? + 1 = 3 2 x 2? + 2 =10 2 x 3? + 3 = 21 I have noticed that each of these results is n less than the number I was aiming to get.

  2. Beyond Pythagoras .

    These formulas are the original formulas for the triples with odd smallest sides. By looking at the table on the previous page, I also noticed something with the 'b' set of numbers. I looked for many paterns and I found out that, 4 X (Triangular number)

  1. Investigate the area of triangle studies including the Pythagorean Theorem and in particular Pythagorean ...

    Middle Side = 112, Largest Side =113. 172 = Middle number + Largest number 289 = Middle number + Largest number 289 = 144.5 2 Lower bound = 144, Upper bound = 145 Middle Side = 144, Largest Side =145. 192 = Middle number + Largest number 361 = Middle number + Largest number 361 = 180.5 2 Lower bound = 180, Upper bound = 181.

  2. Investigating families of Pythagorean triples.

    x ( a + b+ c ) = a ( a + b+ c ) + b ( a + b+ c ) + c ( a + b+ c ) = a2 + b2 + c2 + 2ab + 2ac + 2bc Left Hand Side: ( 2n + 1

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