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Transcript
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 2 Hypotenuse 5 5 cm 3 3 cm 4 cm 4 2 2 2 2 5 =3 + 4 25 = 9 + 16 Opposite the right angle Always the longest side Quit 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 • They wanted an ordered world of real numbers. This length appeared evil to them. 1 Quit • To their horror, the Pythagoreans proved the length of the hypotenuse of this triangle was not a fraction! • 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| |4| + |2| = 90° Quit |3| = 90° z 2 3 4 y z z x y x x Area of square = z2 1 Area of triangle = xy 2 1 Total area = z2 + 4 xy 2 = z2 + 2xy But y Total area = (xz+ y)2 x y = x2 + 2xy + y2 2 Quit x 2 z + 2xy = x + 2xy + y2 x z z z ×4 = (x + y)(x + y) y 2 2 =x +y 2 y z z x y x Do you want to end show? Yes No