# Investigate the area of triangle studies including the Pythagorean Theorem and in particular Pythagorean Triples, sets of numbers where the shortest side is an odd value and all three are positive whole integers.

## The aim of this piece of my coursework is to investigate the area of triangle studies including the Pythagorean Theorem and in particular Pythagorean Triples, sets of numbers where the shortest side is an odd value and all three are positive whole integers. I will collate the relevant data and formulae for the nth terms by using a grid and from this data I should be able to make predictions on the nth terms of Pythagorean Triples. I will keep a narrative of what I am doing and discovering, referring within it to each process of my investigation.

The numbers 3, 4 and 5 satisfy the condition 32+42=52,

Because 32= 3x3 =9
42= 4x4 =16
52= 5x5 =25

And so… 32+42=9+16=25=52

I now have to find out if the following sets of numbers satisfy a similar condition of (smallest number) 2+ (middle number) 2= (largest number) 2.

• 5, 12, 13

52+122 = 25+144 = 169 = 132.

• 7, 24, 25

72+242 = 49+576 = 625 +252

Here is a table containing the results:

I looked at the table and noticed that there was only a difference of 1 between the length of the middle side and the length of the longest side.
I already know that the (smallest number)² + (middle number)² = (largest number)² So, therefore, I know that there will be a connection between the numbers written above. The problem is that it is obviously not:

(Middle number)² + (largest number)² = (smallest number)²

Because: 122 + 132 = 144+169 = 313
52 = 25

The difference between 25 and 313 is 288 which is obviously far too large to be the number I am looking for. This means that the equation I want has nothing to do with 3 sides squared, and I can eliminate this from my list of sides to investigate. I will now try 2 sides squared.

(Middle)² + Largest number = (smallest number)²
= 122 + 13 = 52
= 144 + 13 = 25
= 157 = 25

This does not work and I know that neither will 132, because it is larger than 122. I have decided that there is also no point in squaring the largest and the smallest or the middle number and the largest number. According to the process of elimination and trying all sides to investigate all possible paths, I must now try 1 side squared.

122 + 13 = 5

This couldn’t possibly work, as 122 is already larger than 5, this also goes for 132. The only number now I can try squaring is the smallest number.

12 + 13 = 52
25 = 25

I have finally found the right number - this works with 5 being the smallest number/side but I also need to know if it works with the other 2 triangles I know. I will now square it with the other triples.

4 +5 = 32
9 = 9

24 + 25 = 72
49 = 49

It works with both of my other triangles. So now I know that:

Middle number + Largest number = Smallest number²

If I now work backwards, I should be able to work out some other odd numbers.

i.e  Middle number + Largest number
81 = Middle number + Largest number

I know that there will be only a difference of one between the middle number and the largest number. So, the easiest way to get 2 numbers with only 1 between them is to divide 81 by 2 and then using the upper and lower bound of this number.

81 = 40.5
2

Lower bound = 40, Upper bound = 41.
Middle side = 40, Largest side = 41.

Key

Longest/Largest Side = Length of Longest Side.
Middle Side = Length of Middle Side.
Shortest Side = Length of Shortest Side.

I will do a small investigation to find the number that I already ...