4
8
12 4
12
24 4
16
40 4
20
60
I have found that the middle value has a difference of four. All of the middle values are devisable by 4; I’m going to divide all of middle values by four, then see what further patterns I can spot:
4 4 = 1
12 4 = 3
24 4 = 6
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
I will now find the formula for the shortest side value. I know that the shortest side value is always odd. So I can interpret that the formula would have plus 1, or minus 1 in it. I will list the values as I did for the middle side:
3
2
5
2
7
2
9
Just as I did the middle value I’m going to divide the shortest side value by it’s difference.
3 2 = 1.5
5 2 = 2.5
7 2 = 3.5
9 2 = 4.5
There is a distinct pattern here. I can tell by this that the formula would have 2n in it. I have finally gathered enough information to make a formula.
Formula of the Shortest Side Value
2n + 1
This is what I believe the formula is, I will check it:
N = 4
2 times 4 +1
8 + 1
9
The formula is correct. I have now got the formula for all sides; all I need to do now is figure out the formula for Perimeter and Area.
Formula of Perimeter
I do not have to examine the perimeter’s patterns because the perimeter is all of the sides of triangle added together. I have to add the formulas for each side of a Pythagorean triple together; this will give me the formula for the perimeter.
The longest and Middle sides’ values;
This would be the middle side value times two plus one.
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.
