From the above table, I can see a few patterns emerging. Here they are:

- The shortest side is always an odd number.
- The medium side is always an even number.
- The medium side plus one equals to the Longest side.
- The Longest side is always odd.

To show that these patterns are correct and to see if there are any more patterns, I am going to extend this investigation and draw two more Pythagorean Triples. Here they are:

Again, the above are Pythagorean Triples because they satisfy the condition and all their sides have a positive integer.

Now, I will draw a table showing the results of the first 5 Pythagorean Triples.

As you can see from the above table, the patterns go on and not only that, another pattern has emerged! This is that the Medium side goes up in multiples of 4 each time.

From now on, I am going to abbreviate each quantity and give it a symbol to make it easier for me. Here is a table of the quantities and the symbols that I am going to give them:

There are two types of sequences, Linear sequences and Quadratic sequences. A Linear sequence is when the difference between each number is constant, but a Quadratic sequence is when only the second difference between the numbers is constant. S is a Linear sequence:

However, M and L are Quadratic sequences:

This is the nth term formula to find out S:

This is the nth term formula to find out the M:

As it has been proved earlier that M+1=L, the formula for L will be the same as M plus 1:

Here are the formulas for each side:

I am now going to generate 3 more Pythagorean Triples using the formulae above.

Here is a table showing the results of the 8 Pythagorean Triples:

I am now going to show by algebraic manipulation, that S2+M2=L2:

I am now going to extend this investigation even further and see if I can create other Pythagorean Triplet Families using a source that I have found on the Internet. From the source, I have learnt that other Pythagorean Triplet Families can simply be generated by using a using a different scale, e.g., scale 3, 4, 5 by 2 gives 6, 8, 10 which fits the formula a2+b2=c2 in the same way as scale 3, 4, 5 by 3 gives 9, 12, 15 and fits the formula a+2b2=c2.

I am going to work with scale 2 because I want to prove that other families can indeed be created by simply using a different scale. Here are the first 3 Pythagorean Triples in this family:

I will now work out the Perimeter and Area for the above triangles.

The triangles 1), 2) and 3) are all Pythagorean Triples because they satisfy the condition and all their sides have a positive integers.

Here is a table showing the results of the 3 Pythagorean Triples:

From the above table, I can see a few patterns emerging. Here they are:

- S is always even and goes up in multiples of 4.
- M is always even.
- L is always even.
- M+2=L.

To show that these patterns are correct and to see if there are any more patterns, I am going to extend this investigation and draw two more Pythagorean Triples. Here they are:

Again, the above are Pythagorean Triples because they satisfy the condition and all their sides have a positive integer.

Now, I will draw a table showing the results of the first 5 Pythagorean Triples.

As you can see from the above table, the patterns go on and not only that, other patterns have emerged. This is that the difference between M goes up in multiples of 8 each time, which is double to the multiples of 4 (in the difference), it went up by in the previous set. Also, the Perimeters for these triangles are double to the previous set. I thought this would happen because I doubled all the three sides and so obviously, the Perimeters would double!

Again, S is a Linear sequence and M and L are Quadratic sequences:

This is the nth term formula to find out S:

This is the nth term formula to find out the M:

As it has been proved earlier that M+2=L, the formula for L will be the same as M plus 2:

Here are the formulas for each side:

From this investigation I have learnt that other Pythagorean Triplet Families can be generated by using a different scale and that there are a lot of other families waiting to be investigated!