Introduction Pythagoras Proof of Theorem

2. Introduction Pythagoras Proof of Theorem

3. 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

4. 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.

5. 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

6. 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°

7. 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