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Explore the life of Pythagoras and his theorem: In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. Unravel the mystery and significance of this fundamental concept in mathematics. This secret society, Pythagorean Brotherhood, delved into the mystical world of numbers and philosophy, ultimately shaping the way we understand geometry today.
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Introduction Pythagoras Proof of Theorem
In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides 52 32 42 Hypotenuse 52=32 +42 5 cm 3 cm 25 =9 +16 4 cm Opposite the right angle Always the longest side
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.
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. ? 1 1
Corresponding angles of congruent triangles Angle sum of triangle = 180º x y Construction: 1 Draw a square with sides of length x+y. x z 2 Draw 4 congruent triangles with sides of length x, y, z. y z 3 Label angles 1, 2, 3 and 4 4 Right-angle Proof: y z |1| + |2| = 90° z |1| = |4| x |4| + |2| = 90° y x |3| = 90°
Area of triangle = xy + 4xy 1 2 1 2 x y Area of square = z2 x z Total area =z2 y z z2=x2+y2 =z2 + 2xy × 4 But y z y z Total area = (x + y)2 z = (x + y)(x + y) x =x2+2xy + y2 x y z2 + 2xy =x2+2xy + y2 x
