Pythagoras Theorem The Man and the Theorem

Who was Pythagoras? Pythagoras was born on the Greek island of Samos in c. 475 BC He travelled to Egypt to learn mathematics and astronomy. A Greek coin showing Pythagoras Founded a school in Samos called the Semicircle.

Who was Pythagoras? He founded a secret sect in Croton (Southern Italy) Women were allowed to join this sect A Greek stamp showing Pythagoras The members were vegetarians but beans were excluded from their diet

Who discovered the theorem? Clay tablets (1800 BC and 1650 BC) show that the Babylonians already knew about the Theorem The Egyptians could have used it to construct right angles when they build the pyramids A Babylonian tablet

So what has Pythagoras to do with it? Pythagoras was probably the first to prove the theorem. He is reputed to have proved the theorem while hiding in a cave from the tyrant Polycrates. The cave of Pythagoras at the foot of Mount Kerki, in Samos. Legend has it that he sacrificed an ox to thank the gods!

The Theorem The theorem states that in any right angled triangle …. c b c a The square of the hypotenuse is equal to the sum of the squares of the other sides.

The Theorem b c a c2 = a2 + b2

Proving Pythagoras Theorem There are nearly 400 proofs of the theorem! Among them is a proof by an American president. James A. Garfield 30th President of the United States

Garfield's Proof b a b c a Area of red triangle: ½ a b Area of blue triangle: ½ a b Area of green triangle: ½ c 2 Area of trapezium: ½ (a + b)2 b a

Garfield's Proof a 2 + b 2 = c 2 b a b c a Therefore ½ (a + b)2 = ½ a b + ½ a b + ½ c 2 c (a + b)2 = 2a b + c 2 a 2 + b 2 + 2a b = 2a b + c 2 b a a 2 + b 2 = c 2

Pythagoras' Proof Area of square = c 2 Area of each triangle = ½ ab Area of central square = ½ (a - b) 2

c 2 = a 2 + b 2 Pythagoras' Proof Area of square = c 2 = 4x ½ ab + (a - b)2 = 2ab + a 2 - 2ab + b 2 c 2 = a 2 + b 2

Consider a cuboid of length a, width b and height c. The Theorem in 3-D Consider a cuboid of length a, width b and height c. a b c

We want to find the distance d from one corner to the other The Theorem in 3-D We want to find the distance d from one corner to the other d 2 = x2 + c2 hence d 2 = a2 + b2 + c2 x2 = a2 + b2 d a b c x

The numbers 3, 4 and 5 are said to form a Pythagorean Triple Pythagorean Triples There are cases when the lengths of the sides of a right-angled triangle have integral values 3 4 5 The 3, 4, 5 right-angled triangle is such a case The numbers 3, 4 and 5 are said to form a Pythagorean Triple 52 = 32 + 42

Pythagorean Triples 5 12 13 There are an infinite number of Pythagorean triples Here are two more examples … 25 7 24

finis