Quit Introduction Pythagoras Proof of Theorem Quit 5 2 = 3 2 + 4 2 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the.

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Quit

Introduction Pythagoras Proof of Theorem

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

Quit 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. Pythagoras

Quit 1 1 ? 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.

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

Quit xy x y x x y y z z z z Area of square = z 2 x y Area of triangle = xy 1212 × 4 Total area = z xy 1212 = z 2 + 2xy But Total area = (x + y) 2 = (x + y)(x + y) = x 2 + 2xy + y 2 z 2 + 2xy= x 2 + 2xy + y 2  z 2 = x 2 + y 2 z

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