# Pythagoras.

## Presentation on theme: "Pythagoras."— Presentation transcript:

Pythagoras

Menu Brief History A Pythagorean Puzzle Pythagoras’ Theorem
Using Pythagoras’ Theorem Finding the shorter side Further examples

Pythagoras (~ 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!

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

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

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

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

A Pythagorean Puzzle © R Glen 2001

A Pythagorean Puzzle © R Glen 2001

A Pythagorean Puzzle 1 © R Glen 2001

A Pythagorean Puzzle 1 2 © R Glen 2001

A Pythagorean Puzzle 1 2 © R Glen 2001

A Pythagorean Puzzle 1 2 3 © R Glen 2001

A Pythagorean Puzzle 1 2 3 © R Glen 2001

A Pythagorean Puzzle 1 3 2 4 © R Glen 2001

A Pythagorean Puzzle 1 3 2 4 © R Glen 2001

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

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

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

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

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

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

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

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

A Pythagorean Puzzle x²=y²+z² x² y²
This is called Pythagoras’ Theorem. © 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: hypotenuse

Pythagoras’ Theorem x y z x²=y²+z²

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

Using Pythagoras’ Theorem
What is the length of the slope?

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

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.)

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

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.)

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

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

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.)

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.)

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.)

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

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

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

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