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Introduction

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.

a) 5, 12, 13

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

b) 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 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 far to big, so this means that the equation I want has nothing to do with 3 sides squared. 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 neither will 132, because it is larger than 122. There is also no point in squaring the largest and the smallest or the middle number and the largest number. I will now try 1 side squared.

122 + 13 = 5

This couldn´t work because 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

This works with 5 being the smallest number/side but I need to know if it works with the other 2 triangles I know.

4 +5 = 32
9 = 9

And…

24 + 25 = 72
49 = 49

It works with both of my other triangles. So…

Middle

4 = 4

My formula works for the first term. I will now check if it works using the 2nd term.

4 x 22 - 4(2-1)2 = 12
4 x 4 - 4 x 12 = 12
16 - 4 x 1 = 12
16 - 4 = 12
12 = 12

My formula also works for the 2nd term. It´s looking likely that this is the correct formula. Just to check, I will check if it works using the 3rd term.
4 x 32 - 4(3-1)2 = 24
4 x 4 - 4 x 22 = 24
36 - 4 x 4 = 24
36 – 16 = 24
20 = 24
My formula doesn´t work for the 3rd term. It now looks as if “4n2 - 4(n - 1)2” is not the correct formula after all. To check, I will look to see if the formula works using the 4th term.

4 x 42 - 4(4-1)2 = 40
4 x 16 - 4 x 32 = 40
64 - 4 x 9 = 40
64 – 36 = 40
28 = 40

My formula doesn´t work for the 4th term either. I can now safely say that 4n2 - 4(n-1)2 is definitely not the correct formula for the middle side.
I believe the problem with 4n2 - 4(n-1)2 was that 4n2, once you start using larger numbers, becomes far to high to bring it back down to the number that I want for the middle side. Also, 4(n-1)2 is not as small when it gets larger so it doesn´t bring the 4n2 down enough, to equal the middle side.
I know that the final formula will have something to do with 4 and have to be n2. I will now try n2 + 4.

I will now look at the differences to see if I can find a pattern there.

1 , 4 , 11 , 20 , 31
3 7 9 11
2 2 2

Conclusion

2n2 - 1 and the Relationship between the shortest and middle side both match. So…

Relationship between Shortest and Middle sides = 2n2 – 1

The next relationship I´m going to work out is the relationship between the shortest and longest sides.

Longest Side – Short Side = Relationship.
2n2 + 2n + 1 – 2n + 1 = Relationship.
2n2 + 1 = Relationship.

To check if this formula is right, I am going to write out a table containing 2n2 + 1 and the relationship between the shortest and Longest side.

2n2 + 1 and the Relationship between the shortest and longest sides both match. So…

Relationship between the shortest and longest sides = 2n2 + 1

The next relationship I´m going to find is between the middle and longest side.

Longest Side – middle Side = Relationship.
2n2 + 2n + 1 – 2n2 + 2n = Relationship.
1 = Relationship.

I am certain that the relationship between the longest and middle side is 1, but here´s a table to prove it.

1 and the Relationship between the middle and longest sides both match. So…

Relationship between the middle and longest sides = 1

Using these same principles, I can work out any relationship, to prove this, I will work out the relationship between the shortest side and the perimeter and then the area.

Perimeter - Shortest Side = Relationship.
4n2 + 6n + 2 – 2n + 1 = Relationship.
4n2 + 4n + 1 = Relationship.

I am certain that this is the write answer. So…

Relationship between the Perimeter and Shortest side = 4n2 + 4n + 1

The area is even simpler because, all you have to do is knock the 2n + 1 out of the equation. So…

Relationship between the Area and Shortest side = 2n2 +2n
2
Here is the final table.

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

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