Quit Introduction Pythagoras Proof of Theorem Quit In

Quit

Introduction Pythagoras Proof of Theorem Quit

In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides 5 2 Hypotenuse 5 cm 3 3 cm 4 Quit 2 2 5 =3 + 4 25 = 9 + 16 2 2 Opposite the right angle Always the longest side 2

Pythagoras • Pythagoras lived in the sixth century BC. • He travelled the world to discover all that was known about Mathematics at that time. • He eventually set up the Pythagorean Brotherhood – a secret society which worshipped, among other things, numbers. • Pythagoras described himself as a philosopher – a person whose interest in life is to search for wisdom. Quit

? 1 1 Quit • To their horror, the Pythagoreans proved the length of the hypotenuse of this triangle was not a fraction! • They wanted an ordered world of real numbers. This length appeared evil to them. • Hippasus of Metapontium who leaked the story was thrown out of a boat to drown for threatening the purity of number.

Angle sum of triangle = 180º x Construction: y 1 Draw a square with sides of length x + y. Corresponding angles Draw 4 congruent triangles of congruent triangles y with sides of length x, y, z. z Label angles 1, 2, 3 and 4 Right-angle Proof: | 1| + | 2| = 90° | 1| = | 4| + | 2| = 90° Quit | 3| = 90° z 2 3 4 y z z x y x x

Area of square = z 2 1 Area of triangle = xy 2 1 Total area = z 2 + 4 xy 2 = z 2 + 2 xy But y Total area = (xz + y)2 x y × 4 = (x + y) = x 2 + 2 xy + y 2 Quit 2 x 2 z + 2 xy = x + 2 xy + y 2 y x z z z 2 =x +y 2 2 y z z x y x

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