Transcript Int 1 Unit 2 Pythagoras

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Brief History
A Pythagorean Puzzle
Pythagoras’ Theorem
Using Pythagoras’ Theorem
Finding the shorter side
Further examples
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.
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.
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!
A Pythagorean Puzzle
A right angled triangle
Ask for the worksheet and try
this for yourself!
© R Glen 2001
A Pythagorean Puzzle
Draw a square on
each side.
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A Pythagorean Puzzle
Measure the length
of each side
y
x
z
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A Pythagorean Puzzle
Work out the area of
each square.
y²
y
x²
x
z
z²
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A Pythagorean Puzzle
x²
y²
z²
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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?
1
2
5
3
4
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.
© R Glen 2001
A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
1
5
3
4
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A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
5
1
3
4
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A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
1
3
5
4
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A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
1
5
4
3
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A Pythagorean Puzzle
Put the pieces back
where they came
from.
1
5
4
3
2
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A Pythagorean Puzzle
Put the pieces back
where they came
from.
5
4
2
3
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A Pythagorean Puzzle
x²
y²
This is called
Pythagoras’
Theorem.
x²=y²+z²
z²
© R Glen 2001
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:
Pythagoras’ Theorem
x
y
z
x²=y²+z²
Pythagoras’ Theorem
x
z
x
y
x²=y²+z²
z
y
x
z
y
z
y
x
Using Pythagoras’ Theorem
1m
8m
What is the length of the slope?
Using Pythagoras’ Theorem
y= 1m
x
z= 8m
x²=y²+ z²
x²=1²+ 8²
x²=1 + 64
x²=65
?
Using Pythagoras’ Theorem
x²=65
How do we find x?
We need to use the
square root button on the calculator.
It looks like this
Press
√
√
, Enter 65
So x= √65 = 8.1 m (1
=
Example 1
x
9cm
z
12cm
y
x²=y²+ z²
x²=12²+ 9²
x²=144 + 81
x²= 225
x = √225= 15cm
Example 2
y
z
4m
6m
xs
x²=y²+ z²
s²=4²+ 6²
s²=16 + 36
s²= 52
s = √52
=7.2m (1 d.p.)
Now try Exercise 1
Questions 1 to 10
only!
Finding the shorter side
y
h
7m x
5m
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!
49 - 25 = h²
h²= 24
h = √24 = 4.9 m (1 d.p.)
Example 1
x 13m
x²= y²+ z²
6m
z
w
y
13²= w²+ 6²
169 = w² + 36
169 – 36 = w²
w²= 133Change side,
change sign!
w = √133 = 11.5m
(1 d.p.)
Example 2
z
P
11cm
x
Q
y
9cm
x²= y²+ z²
11²= 9²+ PQ²
121 = 81 + PQ²
121 – 81 = PQ²
R
PQ²=
Change side,
40change sign!
PQ = √40 =
6.3cm
(1 d.p.)
Now try Exercise 1
Questions 11 to 20
Example 1
r
5m
x²=y²+ z²
r²=5²+ 7²
14m
x
r
5m
7m
?
y
r²=25 + 49
½ of 14 r²= 74
z
r = √74
=8.6m (1 d.p.)
38cm
p
23cm
x
y
38cm
23cm
z
Example 2
x²= y²+ z²
38²= y²+ 23²
1444 = y²++ 529
1444 – 529 = y²
y²= 915
Change side,
y = √915=30.2
change sign!
So p =2 x 30.2
= 60.4cm
Now try Exercise 2
Questions 1 to 5