Pythagoras' Theorem

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St. Kentigerns Academy Mathematics
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St. Kentigerns Academy
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Menu
Brief History
A Pythagorean Puzzle
Pythagoras Theorem
Using Pythagoras Theorem
Finding the shorter side
Further examples
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Pythagoras (560-480 B.C.)
Pythagoras was a Greek philosopher and religious
leader.
He was responsible for many important
developments in maths,
astronomy,
and music.
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The Secret Brotherhood
His students formed a secret society called the
Pythagoreans.
As well as studying maths, they were a political
and religious organisation.
Members could be identified by a five pointed
star they wore on their clothes.
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The Secret Brotherhood
They had to follow some unusual rules. They were
not allowed to wear wool, drink wine or pick up
anything they had dropped!

Eating beans was also strictly forbidden!
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A Pythagorean Puzzle
A right angled triangle
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A Pythagorean Puzzle
Draw a square on each side.
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A Pythagorean Puzzle
Measure the length of each side
c
b
a
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A Pythagorean Puzzle
Work out the area of each square.
C²
c
b
b²
a
a²
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A Pythagorean Puzzle
c²
b²
a²
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A Pythagorean Puzzle
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A Pythagorean Puzzle
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A Pythagorean Puzzle
2
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A Pythagorean Puzzle
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A Pythagorean Puzzle
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A Pythagorean Puzzle
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A Pythagorean Puzzle
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A Pythagorean Puzzle
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A Pythagorean Puzzle
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A Pythagorean Puzzle
What does this tell you about the areas of the
three squares?
The red square and the yellow square together
cover the green square exactly.
The square on the longest side is equal in area
to the sum of the squares on the other two sides.
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A Pythagorean Puzzle
Put the pieces back where they came from.
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A Pythagorean Puzzle
Put the pieces back where they came from.
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A Pythagorean Puzzle
Put the pieces back where they came from.
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A Pythagorean Puzzle
Put the pieces back where they came from.
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A Pythagorean Puzzle
Put the pieces back where they came from.
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A Pythagorean Puzzle
Put the pieces back where they came from.
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A Pythagorean Puzzle
c²
b²
c²a²b²
This is called Pythagoras Theorem.
a²
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Pythagoras Theorem
This is the name of Pythagoras most famous
discovery.
It only works with right-angled triangles.
The longest side, which is always opposite the
right-angle, has a special name
hypotenuse
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Pythagoras Theorem
c
a
b
c²a²b²
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Pythagoras Theorem
c
c
b
y
a
a
a
b
b
a
c
c
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Using Pythagoras Theorem
1m
8m
What is the length of the slope?
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Using Pythagoras Theorem
c
a
1m
b
8m
c²a² b²
?
c²1² 8²
c²1 64
c²65
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Using Pythagoras Theorem
c²65
How do we find c?
It looks like this

, Enter 65
Press
So c v65 8.1 m (1 d.p.)
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Example 1
c
c²a² b²
b
c²12² 9²
a
c²144 81
c² 225
c v225 15cm
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Example 2
a
b
c²a² b²
s²4² 6²
c
s²16 36
s² 52
s v52
7.2m (1 d.p.)
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Now try Exercise 4 P156 Then Exercise 5 Problems
involving Pythagoras Theorem
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Finding the shorter side
c
c²a² b²
a
7²a² 5²
49a² 25
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b
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Finding the shorter side
49 a² 25
We need to get a² on its own.
Remember, change side, change sign!
49 - 25 a²
a² 24
a v24 4.9 m (1 d.p.)
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Example 1
c
c² a² b²
13² a² 6²
b
169 w² 36
a
169 36 a²
a² 133
a v133 11.5m (1 d.p.)
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Example 2
c² a² b²
b
11² 9² b²
a
121 81 b²
c
121 81 b²
b² 40
b v40 6.3cm (1 d.p.)
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Now try Exercise 6 P 161
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Example 1
c²a² b²
c²5² 7²
c²25 49
c
c² 74
r
5m
b
c v74
8.6m (1 d.p.)
?
7m
a
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Example 2
c² a² b²
38² a² 23²
1444 a² 529
c
1444 529 y²
38cm
a² 915
a
a v915
So a 2 x v915 60.5cm
(1d.p.)
23cm
b
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Now try Exercise 2 Questions 1 to 5
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St. Kentigerns Academy Mathematics