# 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:

a + b= c

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.

Middle

40 4 = 10

60 4 = 15

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

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 ) )

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 ) )

Now I will simplify the formula:

4 ( n ( n + 1 ) )

Multiply out first bracket.

4n (n + 1) Divide top line by 2

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

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.

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

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