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Math's Coursework: Pythagoras triples.

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Introduction

Math’s Coursework: Pythagoras triples.

        I am investigating the relationship of the sides in a Pythagoras triple. I will hopefully be able to find formulas for each side, and the perimeter and Area. To do this I must first discover the associations each side of a Pythagorean triples have (the shortest, middle and longest side.)

        A Pythagorean triples are basically the sides of a right-angle triangle according to Pythagoras. The formula is:

aimage00.png + bimage00.png= cimage00.png

Length of Shortest side ( a )

Length  of middle side ( b )

Length of longest side  ( c )

Perimeter

Area

3

4

5

12

6

5

12

13

30

30

7

24

25

56

84

        This is a table of the sides, perimeter and area of the first three Pythagorean triples. Judging by the table I can recognize a few of relationships between the numbers:

  • The shortest and longest sides are odd numbers.
  • The value of the middle side is the longest side’s value -1.
  • The lengths of the shortest sides are 2 more then the previous shortest side length.
...read more.

Middle

 4 = 6

  40 image02.png 4 = 10

  60 image02.png 4 = 15

        I have established that the middle values divided by four equal triangular numbers.

image03.png

        I have found a vital pattern. Using the formula I know for triangular numbers I can find a formula for the Pythagorean triples, in consideration of its order.

 Formula of the Middle Side Value

        This is the formula of triangular numbers.    ( n ( n + 1 ) )

image01.png

        The middle side value, because it is simply four times a triangular number, should be the formula above multiplied by four:

                                        4 ( n ( n + 1 ) )image01.png

Now I will simplify the formula:

                4 ( n ( n + 1 ) )image01.png

                                               Multiply out first bracket.

                4n (n + 1)                Divide top line by 2image01.png

                2n (n + 1)

        I have simplified the formula, now to check it:

                                                        N= 4      

                8 ( 4 + 1 )

                8 times 5 = 40

        The formula is correct, with this I know the formula for the longest side, which is just plus 1 of the middle side value.

 Pattern of the Shortest Side Value

...read more.

Conclusion

                2 ( 2n ( n + 1) ) +1

Simplify:                                Multiply out first set of brackets

                4n ( n + 1 ) + 1

        This is as much as I can simplify it. I am going to check it:

                                                        N = 4

                4 times 4 ( 4 + 1 ) + 1

                16 ( 4 + 1 ) + 1

                ( 16 times 5 ) + 1

                80 + 1 = 81

I have to add this formula to the formula for the shortest sides.

                ( 2n +1 ) + ( 4n ( n + 1 ) + 1 )

Check:                                                 n = 4

                ( 8 + 1 ) + ( 16 times 5) + 1 )

                9 + 81 = 90

The formula is correct. My next step is to figure out the formula for the Area.

Formula for Area

 The formula for working out the area of a triangle is:   ½ (bh)

        Just as I did for the perimeter I can use my previous formulas to find a formula for the area of Pyth triples. The formula needs all sides excluding the longest side which is the hypotenuse.

                  (2n + 1)  (  2n ( n + 1 ) )  

                                      2

Simplify                                                                   divide by 2

                (2n + 1)  (n (n + 1)

Check:                                                 n = 4

                (2 times 4 + 1) ( 4 (4 + 1 )

                9 times  ( 4 times 5)

                9 times 20 = 180

I have found all of the formulas for Pythagorean triples, after I found the formula of each side the rest was fairly easy.

...read more.

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