Pythagoras Theorem

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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:

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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 ...

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