# Pythagoras Theorem

Beyond Pythagoras

Pythagoras was a famous Greek mathematician. He produced a formula (a² + b² = c²) called the ‘Pythagoras Theorem.’ This allows you to correctly calculate the length of a right-angled triangles’ hypotenuse, when you know the length of the other two sides.

A Pythagorean triple is when a right-angled triangles three lengths (a = shortest, b = medium, c = hypotenuse) are all whole numbers- integers. At the start of this course, I was given a set of 3 Pythagorean triples. These were:

I was asked to look for patterns between the numbers in each Pythagorean triple. This is what I found:

• The difference between b and c is one
• a was odd
• b was even and a multiple of four
• c was odd
• c = b + 1
• (a² + 1)/2 = c
• (a² - 1)/2 = b

The two formulae that I am going to focus on, and use to discover more Pythagorean Triples are:

• (a² + 1)/2 = c
• (a² - 1)/2 = b

Using these two rules I can use some more odd numbers to expand my table to:

To test that these were possible Pythagorean triples, I put them through the original Pythagoras theory:

• 9² + 40² = 1,681                                √1,681 = 41
• 11² + 60² = 3,721                              √3,721 = 61
• 13² + 84² = 7,225                              √7,225 = 85
• 15² + 112² = 12,769                          √12,769 = 113

I had noticed earlier that ‘a’ was an odd number and I wanted to see if Pythagorean triples would work when ‘a’ was an even number. To make the value of ‘a’ even I decided to double the original three sets of triples:

I then tested this new set of numbers, by putting them trough the original theorem:

• 6² + 8² = 100                                             √100 = 10
• 10² + 24² = 676                                          √676 = 26
• 14² + 48² = 2,500                                        √2,500 ...