St. Kentigerns Academy

Menu 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!

A Pythagorean Puzzle

Draw a square on each side.

A Pythagorean Puzzle

Measure the length of each side b c a

A Pythagorean Puzzle

Work out the area of each square.

b² b c C² a a²

A Pythagorean Puzzle

c² b² a²

A Pythagorean Puzzle

A Pythagorean Puzzle



1

A Pythagorean Puzzle 1



2

A Pythagorean Puzzle 1 2

A Pythagorean Puzzle 1 2 3

A Pythagorean Puzzle 1 2 3

A Pythagorean Puzzle 1 3 2 4

A Pythagorean Puzzle 1 3 2 4

A Pythagorean Puzzle 1 3 2 5 4

A Pythagorean Puzzle

What does this tell you about the areas of the three squares?

2 1 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.

A Pythagorean Puzzle

Put the pieces back where they came from.

2 1 5 3 4

A Pythagorean Puzzle

Put the pieces back where they came from.

2 1 3 5 4

A Pythagorean Puzzle

Put the pieces back where they came from.

2 1 3 5 4

A Pythagorean Puzzle

Put the pieces back where they came from.

2 1 5 3 4

A Pythagorean Puzzle

Put the pieces back where they came from.

1 5 3 4 2

A Pythagorean Puzzle

Put the pieces back where they came from.

5 4 3 1 2

A Pythagorean Puzzle

b² This is called Pythagoras’ Theorem.

a² c²

c²=a²+b²

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

a c b

c²=a²+b²

a a c

Pythagoras’ Theorem

c y

c²=a²+b²

a b b c b a c

Using Pythagoras’ Theorem

1m 8m What is the length of the slope?

Using Pythagoras’ Theorem

c a= 1m b= 8m c²=a²+ b²

?

c²=1²+ 8² c²=1 + 64 c²=65

Using Pythagoras’ Theorem

c²=65 How do we find c?

We need to use the square root button on the calculator.

It looks like this

√

Press

√

, Enter 65

=

So c=

√

65 = 8.1 m (1 d.p.)

Example 1

c c²=a²+ b² c²=12²+ 9² c²=144 + 81 c²= 225 c =

√

225= 15cm

Example 2

b 6m c s c²=a²+ b² s²=4²+ 6² s²=16 + 36 s²= 52 s =

√

52 =7.2m (1 d.p.)

Now try Exercise 4 P156 Then Exercise 5 Problems involving Pythagoras Theorem

a h

Finding the shorter side

7m c c²=a²+ b² 7²=a²+ 5² 49=a² + 25

?

Finding the shorter side

We need to get a² on its own.

Remember, change side, change sign!

49 - 25 = a² a²= 24 a =

√

24 = 4.9 m (1 d.p.)

Example 1

c 13m w a c²= a²+ b² 13²= a²+ 6² 169 – 36 = a² a²= 133 Change side, change sign!

a =

√

133 = 11.5m

(1 d.p.)

P

Example 2

11cm c b Q a 9cm R c²= a²+ b² 11²= 9²+ b² 121 – 81 = b² b²= 40 Change side, change sign!

b =

√

40 = 6.3cm

(1 d.p.)

Now try Exercise 6* P 161

Example 1

r

?

5m c²=a²+ b² c²=5²+ 7² 14m 5m ½ of 14 b c²=25 + 49 c²= 74 c =

√

74 =8.6m (1 d.p.)

p 38cm 23cm a c 38cm

Example 2

c²= a²+ b² 38²= a²+ 23² 1444 = a²+ 529 1444 – 529 = y² a²= 915 a = Change side, change sign!

So a =2 x

√

915 = 60.5cm (1d.p.)

Now try Exercise 2 Questions 1 to 5