Pythagoras Theorem Pythagoras of Samos Born: about 569 BC in Samos, Ionia Died: about 475 BC Pythagoras of Samos gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans. (i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans knew the generalisation which states that a polygon with n sides has sum of interior angles 2n - 4 right angles and sum of exterior angles equal to four right angles. (ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square. (iii) Constructing figures of a given area and geometrical algebra. For example they solved equations such as a (a - x) = x2 by geometrical means. (iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number. (v) The five regular solids. It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two. (vi) In astronomy Pythagoras taught that the Earth was a sphere at the centre of the Universe. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star. Pythagoras believed that all relations could be reduced to number relations

Babylonians’ Theorem! Pythagoras Today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it.

Area = 25 3 5 4 Area = 16 Area = 9 16 + 9 = 25

In general The sum of the areas for the two smaller squares is equal to the area of the larger square . Or in other words the sum of the squares of the two smaller sides is equal to the square of the largest side. Or in algebraic terms The largest side is known as the Hypotenuse b c a

Babylonian Tablet Certainly the Babylonians were familiar with Pythagoras's theorem. A translation of a Babylonian tablet which is preserved in the British museum goes as follows:- 4 is the length and 5 the diagonal. What is the breadth ? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9 ? 3 times 3 is 9. 3 is the breadth. All the tablets we wish to consider in detail come from roughly the same period, namely that of the Old Babylonian Empire which flourished in Mesopotamia between 1900 BC and 1600 BC.

Example Calculate the lettered length in each of the following right angled triangles : (i) c 5 cm 12 cm

(ii) d 17 cm 24 cm

(iv) e 15 cm 8 cm

(ii) 9 cm f 12 cm

Find the length of PQ correct to 2 decimal places Example Find the length of PQ correct to 2 decimal places 21 m 15 m P R Q

14 cm 9 cm y Example Find the value of y correct to 1 decimal place. Pythagorean Triples by supplying two different positive integer values for m and n in this diagram. You can multiply out these terms and check that ( m2 – n2 )2 + (2 m n)2 = ( m2 + n2 )2