Download Chapter 6: Pythagoras` Theorem

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Introduction
Pythagoras
Proof of Theorem
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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
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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.
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1
• They wanted an ordered world of real
numbers. This length appeared evil to
them.
1
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• 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°
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|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
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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
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