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Introduction

Pythagoras

Proof of Theorem

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5 2 = 3

2 + 4

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

3

25 cm

Hypotenuse5

2

4

2

25 = 9 + 16

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• 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.

PythagorasPythagoras

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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.

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Construction:

Draw a square with sidesof length x + y.

x y

xy

x

x

y

y

1

2

34

zz

z

z

Draw 4 congruent triangleswith 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

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x y

xy

x

x

y

y zz

z

z

Area of square = z2

x

y

Area of triangle = xy12

× 4

Total area = z2 + 4 xy12

= z2 + 2xy

But

Total area = (x + y)2

= (x + y)(x + y)

= x2 + 2xy + y2

z2 + 2xy = x2 + 2xy + y2

z 2 = x

2+ y

2

z

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