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

St. Kentigerns Academy

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


Ask for the worksheet and try this for yourself!

Draw a square on each side.


cb

a

Measure the length of each side


Work out the area of each square.


a

b

C²

b²

a²

c


c²

b²

a²


1


1

2


1

2


1

2

3


1

2

3


1

23

4 A Pythagorean Puzzle

1

23

4


1

23

45


1

23

4

5

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.


1

23

4

5

Put the pieces back where they came from.


1

23

45



1

23

4

5



1

2

3

4

5



1

23

4

5



1

23

4

5



This is called Pythagoras’ Theorem.


c²

b²

a²

c²=a²+b²

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.


c

b

a

c²=a²+b²


c

a

c

c

b

b

b a

a

c

ya


c²=a²+b²

1m

8m


What is the length of the slope?

1m

8m

c

b=

a=

c²=a²+ b²

c²=1²+ 8²

c²=1 + 64

c²=65

?


How do we find c?

We need to use the

square root button on the calculator.It looks like this √

Press

c²=65

√ , Enter 65 =

So c= √65 = 8.1 m (1 d.p.)


Example 1

c

12cm

9cm

a

bc²=a²+ b²

c²=12²+ 9²

c²=144 + 81

c²= 225

c = √225= 15cm

c

6m4m

s

ab

c²=a²+ b²

s²=4²+ 6²

s²=16 + 36

s²= 52

s = √52

=7.2m (1 d.p.)

Example 2

Now try Exercise 4P156

Then Exercise 5 Problems involving

Pythagoras Theorem

7m

5m

hc

a

b

c²=a²+ b²

7²=a²+ 5²

49=a² + 25?


49 = a² + 25

We need to get a² on its own.Remember, change side, change sign!


+ 25

49 - 25 = a²

a²= 24

a = √24 = 4.9 m (1 d.p.)

169 = w² + 36

c

w

6m

13m

a

b

c²= a²+ b²

13²= a²+ 6²

169 – 36 = a²

a = √133 = 11.5m (1 d.p.)

a²= 133

Example 1

169 = a² + 36

Change side, change sign!

c

b c²= a²+ b²

11²= 9²+ b²

121 = 81 + b²

121 – 81 = b²

b = √40 = 6.3cm (1 d.p.)

b²= 40

a9cm

P

11cm

R

Q

Example 2

81


Now try Exercise 6*P 161

c

a

b

c²=a²+ b²

c²=5²+ 7²

c²=25 + 49

c²= 74

c = √74

=8.6m (1 d.p.)

14m

5mr

r5m

7m

Example 1

½ of 14

?

c

a

b

23cm

38cm

p

38cm

23cm

c²= a²+ b²

38²= a²+ 23²1444 = a²+ 5291444 – 529 = y²

a = √915

a²= 915

So a =2 x √915 = 60.5cm (1d.p.)

Example 2

+ 529


Now try Exercise 2

Questions 1 to 5

Documents

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