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

 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:

 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.