Beyond Pythagoras.

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Pythagoras Theorem is the formula for working out the hypotenuse of a right-angled triangle when the other two sides of the triangle are given. The formula for Pythagoras Theorem is:

a²+b²=c².

In this formula, a is the shortest side, b is the middle side, and c is the shortest side.

Pythagorean Triples are when the three sides of the right-angled triangle are all whole numbers. The first five Pythagorean Triples are:

3, 4, 5-because 9(3²) +16(4²) =25(5²)

5, 12, 13-because 25(5²) +144(12²) =169(13²)

7, 24, 25-because 49(7²) +576(24²) =625(25²)

9, 40, 41-because 81(9²) +1600(40²) =1681(41²)

1, 60, 61-because 121(11²) +3600(60²) =3721(61²)

As you may have noticed, the last two numbers (b and c) have a difference of 1. You may have also noticed that the first and last numbers (a and c) are odd numbers and the middle numbers (b) are even.

This investigation is about finding a formula for various different purposes including finding the nth Pythagorean Triple, the perimeter of a Pythagorean Triple and the area of a Pythagorean Triple.

Firstly, I will put the values of the first five Pythagorean Triples into table:

Term number (n)

Short side (a)

Middle side (b)

Long side (c)

3

4

5

2

5

2

3

3

7

24

25

4

9

40

41

5

1

60

61

Just by looking at the table, I can work out the formula for finding the short side of a Pythagorean Triple in terms of n. Just in case, I worked it out:

0 1 2 3 4 5

3 5 7 9 11

2 2 2 2 2

I wrote down the first five terms of the sequence, I worked out the difference between the terms, and then, using the difference, I worked out the zeroth term of the formula.

I found that this sequence was a linear sequence as it only had one difference.

The general formula for a linear sequence is bx+c.

To work out b, you have to find the second difference of the sequence.

In this case, it is 2 so b=2.

To work out c, you have to find the zeroth term.

In this case it is 1 so c=1

So the formula for finding the short side of a Pythagorean Triple in terms of n is:

2n+1

Term number (n)

Short side (a)

Middle side (b)

Long side (c)

3

4

5

2

5

2

3

3

7

24

25

4

9

40

41

5

1

60

61

Now, I had to work out the formula for finding the middle side of a Pythagorean Triple in terms of n:

0 1 2 3 4 5

0 4 12 24 40 60

4 8 12 16 20

4 4 4 4

As you can see, this sequence is different to the previous sequence. It has a second difference, so it is termed as a quadratic sequence.
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The general formula for a quadratic sequence is ax²+bx+c.

To work out a, you have to divide the second difference by 2.

In this case, the second difference is 4, and 4÷2=2, so a=2.

You cannot work out b before you have worked out c.

To work out c, you have to find the zeroth term.

In this case, it is 0 so c=0.

The formula I have worked out so far is 2n²+0.

Now I have to work out b.

To work out b, you have to substitute 1 ...

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