Brief History Finding the shorter side A Pythagorean Puzzle Pythagoras’ Theorem Using...
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Brief History
Finding the shorter side
A Pythagorean Puzzle
Pythagoras’ Theorem
Using Pythagoras’ Theorem
Menu
Further examples
Pythagoras was a Greek philosopher and religious leader.He was responsible for many important developments in maths,
astronomy, and
music.
Pythagoras (~560-480 B.C.)
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.
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!
The Secret Brotherhood
A right angled triangle
A Pythagorean Puzzle
Ask for the worksheet and try this for yourself!
© R Glen 2001
Draw a square on each side.
A Pythagorean Puzzle
© R Glen 2001
xy
z
Measure the length of each side
A Pythagorean Puzzle
© R Glen 2001
Work out the area of each square.
A Pythagorean Puzzle
x
z
y
x²
y²
z²
© R Glen 2001
A Pythagorean Puzzle
x²
y²
z²
© R Glen 2001
A Pythagorean Puzzle
© R Glen 2001
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A Pythagorean Puzzle
© R Glen 2001
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A Pythagorean Puzzle
© R Glen 2001
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A Pythagorean Puzzle
© R Glen 2001
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A Pythagorean Puzzle
© R Glen 2001
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A Pythagorean Puzzle
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A Pythagorean Puzzle
© R Glen 2001
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A Pythagorean Puzzle
© R Glen 2001
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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.
A Pythagorean Puzzle
© R Glen 2001
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4
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Put the pieces back where they came from.
A Pythagorean Puzzle
© R Glen 2001
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A Pythagorean Puzzle
Put the pieces back where they came from.
© R Glen 2001
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A Pythagorean Puzzle
Put the pieces back where they came from.
© R Glen 2001
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A Pythagorean Puzzle
Put the pieces back where they came from.
© R Glen 2001
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A Pythagorean Puzzle
Put the pieces back where they came from.
© R Glen 2001
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A Pythagorean Puzzle
Put the pieces back where they came from.
© R Glen 2001
This is called Pythagoras’ Theorem.
A Pythagorean Puzzle
x²
y²
z²
x²=y²+z²
© R Glen 2001
It only works with right-angled triangles.
hypotenuse
The longest side, which is always opposite the right-angle, has a special name:
This is the name of Pythagoras’ most famous discovery.
Pythagoras’ Theorem
x
z
y
x²=y²+z²
Pythagoras’ Theorem
x
y
x
x
y
y
z z
z
x
yz
Pythagoras’ Theorem
x²=y²+z²
1m
8m
Using Pythagoras’ Theorem
What is the length of the slope?
1m
8m
x
z=
y=
x²=y²+ z²
x²=1²+ 8²
x²=1 + 64
x²=65
?
Using Pythagoras’ Theorem
How do we find x?
We need to use the
square root button on the calculator.It looks like this √
Press
x²=65
√ , Enter 65 =
So x= √65 = 8.1 m (1 d.p.)
Using Pythagoras’ Theorem
Example 1
x
12cm
9cm
y
zx²=y²+ z²
x²=12²+ 9²
x²=144 + 81
x²= 225
x = √225= 15cm
x
6m4m
s
yz
x²=y²+ z²
s²=4²+ 6²
s²=16 + 36
s²= 52
s = √52
=7.2m (1 d.p.)
Example 2
What’s in the box?
24cm
7cm
25 cm
7m5m
8.6 m to 1 dp
Problem 1
Problem 2
7m
5m
hx
y
z
x²=y²+ z²
7²=h²+ 5²
49=h² + 25?
Finding the shorter side
49 = h² + 25
We need to get h² on its own.Remember, change side, change sign!
Finding the shorter side
+ 25
49 - 25 = h²
h²= 24
h = √24 = 4.9 m (1 d.p.)
169 = w² + 36
x
w
6m
13m
y
z
x²= y²+ z²
13²= w²+ 6²
169 – 36 = w²
w = √133 = 11.5m (1 d.p.)
w²= 133
Example 1
169 = w² + 36
Change side, change sign!
x
z x²= y²+ z²
11²= 9²+ PQ²
121 = 81 + PQ²
121 – 81 = PQ²
PQ = √40 = 6.3cm (1 d.p.)
PQ²= 40
y9cm
P
11cm
R
Q
Example 2
81
Change side, change sign!
What’s in the box 2?
9cm
A
11cm
C
B
6.3 cm to 1 dp
9m
4.5m
7.8m to 1 dp
Problem 1
Problem 2
x
y
z
x²=y²+ z²
r²=5²+ 7²
r²=25 + 49
r²= 74
r = √74
=8.6m (1 d.p.)
14m
5mr
r5m
7m
Example 1
½ of 14
?
x
y
z
23cm
38cm
p
38cm
23cm
x²= y²+ z²
38²= y²+ 23²1444 = y²+ 5291444 – 529 = y²
y = √915=30.2
y²= 915
So p =2 x 30.2 = 60.4cm
Example 2
+ 529
Change side, change sign!
What’s in the box 3?
20m
8mrr = 12.8m to 1 dp
30cm
42cm
pp = 58.8m to 1 dp
Problem 1
Problem 2