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Pythagoras

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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 (~ 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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**Ask for the worksheet and try this for yourself!**

A Pythagorean Puzzle A right angled triangle Ask for the worksheet and try this for yourself! © R Glen 2001

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A Pythagorean Puzzle Draw a square on each side. © R Glen 2001

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**A Pythagorean Puzzle x y z Measure the length of each side**

© R Glen 2001

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**A Pythagorean Puzzle x y z Work out the area of each square. x² y² z²**

© R Glen 2001

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A Pythagorean Puzzle x² y² z² © R Glen 2001

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A Pythagorean Puzzle © R Glen 2001

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A Pythagorean Puzzle 1 © R Glen 2001

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A Pythagorean Puzzle 1 2 © R Glen 2001

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A Pythagorean Puzzle 1 2 © R Glen 2001

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A Pythagorean Puzzle 1 2 3 © R Glen 2001

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A Pythagorean Puzzle 1 2 3 © R Glen 2001

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A Pythagorean Puzzle 1 3 2 4 © R Glen 2001

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A Pythagorean Puzzle 1 3 2 4 © R Glen 2001

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A Pythagorean Puzzle 1 3 2 5 4 © R Glen 2001

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A Pythagorean Puzzle 1 What does this tell you about the areas of the three squares? 3 2 5 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

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**A Pythagorean Puzzle Put the pieces back where they came from. 1 3 5 2**

4 © R Glen 2001

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**A Pythagorean Puzzle Put the pieces back where they came from. 1 3 2 5**

4 © R Glen 2001

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**A Pythagorean Puzzle Put the pieces back where they came from. 1 3 2 5**

4 © R Glen 2001

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**A Pythagorean Puzzle Put the pieces back where they came from. 1 2 5 4**

3 © R Glen 2001

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**A Pythagorean Puzzle Put the pieces back where they came from. 1 5 4 2**

3 © R Glen 2001

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**A Pythagorean Puzzle Put the pieces back where they came from. 5 4 2 3**

1 © R Glen 2001

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**A Pythagorean Puzzle x²=y²+z² x² y²**

This is called Pythagoras’ Theorem. z² © R Glen 2001

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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 x y z x²=y²+z²

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Pythagoras’ Theorem x x y z x²=y²+z² y z y y z z x x

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**Using Pythagoras’ Theorem**

What is the length of the slope?

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**Using Pythagoras’ Theorem**

x y= 1m z= 8m x²=y²+ z² ? x²=1²+ 8² x²=1 + 64 x²=65

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**Using Pythagoras’ Theorem**

x²=65 We need to use the square root button on the calculator. How do we find x? √ It looks like this √ Press , Enter 65 = So x= √65 = 8.1 m (1 d.p.)

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**Example 1 x z y 9cm x²=y²+ z² x²=12²+ 9² 12cm x²=144 + 81 x²= 225**

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**Example 2 y z x 4m 6m x²=y²+ z² s²=4²+ 6² s s²=16 + 36 s²= 52 s = √52**

=7.2m (1 d.p.)

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**What’s in the box? Problem 1 Problem 2 7cm 24cm 5m 7m 25 cm**

8.6 m to 1 dp

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**Finding the shorter side**

7m 5m h x x²=y²+ z² y 7²=h²+ 5² 49=h² + 25 ? z

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**Finding the shorter side**

+ 25 We need to get h² on its own. Remember, change side, change sign! = h² h²= 24 h = √24 = 4.9 m (1 d.p.)

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**Example 1 x z y w 6m 13m x²= y²+ z² 13²= w²+ 6² 169 = w² + 36**

Change side, change sign! y 169 – 36 = w² w²= 133 w = √133 = 11.5m (1 d.p.)

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**Example 2 z y x x²= y²+ z² 9cm P 11cm R Q 11²= 9²+ PQ² 81**

Change side, change sign! 121 = 81 + PQ² x 121 – 81 = PQ² PQ²= 40 PQ = √40 = 6.3cm (1 d.p.)

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**What’s in the box 2? Problem 1 9m 4.5m Problem 2 A B 9cm 11cm**

6.3 cm to 1 dp 9cm A 11cm C B 9m 4.5m 7.8m to 1 dp Problem 2

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**Example 1 x z ? y r 5m x²=y²+ z² r²=5²+ 7² 14m r²=25 + 49 r r²= 74**

½ of 14 r²= x r r²= 74 5m z r = √74 =8.6m (1 d.p.) 7m ? y

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**Example 2 x y z 23cm 38cm p x²= y²+ z² 38²= y²+ 23² 1444 = y²+ 529**

Change side, change sign! x 1444 – 529 = y² 38cm y²= 915 y y = √915=30.2 So p =2 x 30.2 = 60.4cm 23cm z

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**What’s in the box 3? Problem 1 Problem 2 20m 8m r 12.8m to 1 dp r =**

30cm 42cm p p = 58.8m to 1 dp Problem 2

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