1. The Theorem of Pythagoras

3. The longest side, opposite the right angle, is called the hypotenuse
Pythagoras’ Theorem
In a right angled triangle:
c2 = a2 + b2 a b c
The longest side, opposite the right angle, is called the hypotenuse

4. Find x in the diagram below
Pythagoras’ Theorem: c2 = a2 + b2
15 cm
22 cm x
Longest side (hypotenuse) = x
So x2 = =
= 709
To find x we find the
square root of 709
So x = 709
= 26•63

5. Worked Example 5 x 9

6. Worked Example 2 4 x 6

7. 1 2 3 4 Pythagoras’ Theorem: c2 = a2 + b2 Find x in each diagram below
12 cm x
10·5 cm 2 x
3 cm 1
8 cm x
9·1 m
15·4 m 3 x
72 mm
43 mm 4

8. Pythagoras’ Theorem: c2 = a2 + b2
Now we’ll see what happens when we need to look for a shorter side
Find x
11 cm
27 cm y
Pythagoras’ Theorem: c2 = a2 + b2
Longest side (hypotenuse) = 27 (known)
x2 =
When we wanted to find the longest side we added the squares. =
= 608
To find a shorter side we subtract the squares.
x = 608
Longest – ADD
Shorter - SUBTRACT
= 24•66

9. 8 x 11

10. 6.7 x
3.5

11. 2 1 3 4 Calculate the value of ‘x’ in each of the examples below x mm
12 cm
x cm 2 1
43 mm
9·75 cm
85 mm
7 cm
15·1 cm
3·1 m
9·7 cm
1·62 m 3 4
x m
3·5 m
11·6 cm
x cm

12. Try these examples x 8 4 x 5 5 3 x x 14 18 13 x 6 x
6.7
4.1

13. Now we’ll look at two worded problems which involve the use of Pythagoras

14. ( 4•2 – 3) This house extension has a roof as shown.
Calculate the width of the extension.
3 m
4•2 m
4 m
Width
extract the right-angled triangle
write in the lengths of each side 4
( 4•2 – 3)
1•2 w
w2 = •22
= •44
= 14•56
x = 14•56
= 3•82

15. A flagpole is supported by 2 ropes as shown in the diagram
A flagpole is supported by 2 ropes as shown in the diagram. Find the height of the flagpole.
25 m
14 m
1 m
extract the right-angled triangle
write in the lengths of each side h
25 m
7 m
half of 14 m
h2 = =
= 576
h = 576
The flagpole = = 25 m
= 24

16. Pythagoras
Problems

17. A ship leaves port and sails 18km north.
It then turns and sails 15km west.
How far is the ship from the port?
The hands of a clock are 8.4cm and 5.8cm long.
How far apart are the tips of the hands at 9pm?

18. Calculate the value of x in the
diagram shown. Give your answer correct to 1 decimal place.
6cm x
11cm
16cm
19cm
The diagram shows a rhombus with
diagonals 19cm and 9cm long.
Calculate the length of the side
of the rhombus.
9cm

19. x On square paper plot the points A(-4,2), B(7,-3) and C(3,6)
and form a triangle.
Use Pythagoras to find the perimeter of the triangle. x
14m
wall
ladder 6m
ground 9m
A wall is 6m high.
A ladder 14m long rests against the wall with the foot of
the ladder 9m from the bottom of the wall as shown in the diagram.
What length of the ladder x hangs over the top of the wall?

20. Use Pythagoras to find the length of the line marked4 10 x
Calculate the values of x
and y in the diagram shown. 8 y
Use Pythagoras to find the
length of the line marked
with an x. 14 5 9 x

21. If A(1,2), B(5,1) and C(4,-3) are 3 vertices of a
rectangle ABCD, find the coordinates of D.
Draw the rectangle on square paper.
Use Pythagoras to find the lengths of the sides and
the diagonals of the rectangle.
An aeroplane leaves an airport and flies
98 km north east. It then turns and flies
114 km south east.
How far is the aeroplane from the airport?

22. An equilateral triangle has Find the length of the dotted
side of length 8cm.
Find the length of the dotted
line in the diagram. 6 B 5 3
Find the perimeter of
triangle ABC. A
5.4 9 C
2.8

23. The size of a television screen is given as the size of the diagonal length of the screen.
If a television screen has a length of 32 inches and a breadth of 27 inches, what size screen does the television have?
70m
The diagram shows an arena
used for animals at the
Royal Agricultural Show
at Ingliston.
Calculate the total length of
fencing needed to construct
this arena.
25m
goats
pigs
16m
42m
cows
sheep
35m

24. The lengths of the diagonals of a kite are 16cm and 6cm respectively.
Calculate the lengths of the
sides of the kite
10cm
6 cm
16cm
Which of the points A(4,5), B(3,-5)
and C(-6,2) is furthest away from the origin?

25. Use Pythagoras to find x.
6cm
4cm
Use Pythagoras to find x.
3cm
12cm
Calculate the length of the side
marked x in the diagram shown.
Give your answer correct
to 1 decimal place.
6cm x
5cm
12cm

26. An aero plane flight BA2184 leaves an airport and flies 90 miles north and then 80 miles west.
At the same time a second aeroplane Ryan4456 leaves the airport and flies at the same speed, 60 miles south and then 110 miles east.
How far apart are the two planes at this time?
A circle has a centre C(5,2). The point A(8,6) lies on the circumference of the circle.
Calculate the radius of the circle.
Which of the points P(9,0), Q(2,6) and R(1,1) lie on the circle.

27. 12 x 2x
The diagram shows a right angled triangle.
Calculate the lengths of the sides of the triangle.

28. Find the distance between the points (1,2) and (7,9)x y
By Pythagoras:
(7,9)
d 2 = ●
d 2 = d 7
d 2 = 85
d = √85 ● 6
(1,2)
d = 9·22

29. Find the distance between the points (-5,-2) and (6,7)x y
By Pythagoras:
d 2 =
d 2 = ●
(6,7)
d 2 = 202 d
d = √202 9
d = 14·2 ● 11
(-5,-2)

30. The Converse Of Pythagoras.
14m
19m
23m
Is the triangle right angled or not ?
©Microsoft Word clipart

31. What Is A Converse ? Consider the sentence below: If
the angles of the shape add up to 180o
Then
the shape is a triangle.
To make the converse statement swap around the parts of the statement in the white box: If
the shape is a triangle.
Then
the angles of the shape add up to 180o
This is the converse statement.

32. Not all converse statements are true.
Consider the sentence below: If
a shape is a square
Then
the angles add up to 360o
Now make the converse statement. If
the angles add up to 360o
Then
a shape is a square.
Can you think of a shape with angles of 360o which is not a square ?
Any closed quadrilateral.

33. What Goes In The Box ?
Write the converse statement and decide if the converse of the statement is true or false.
(1) If a shape is a square then the shape has parallel sides.
False
(2) If a number is even then the number divides by two exactly.
True
(3) If you have thrown a double six with a dice then your score with the dice is twelve.
True
(4) If you have thrown a three and a four then your total score is seven with a dice.
False

34. The Converse Of Pythagoras.
The Theorem Of Pythagoras states: If
a given triangle is right angled
Then
a2 + b2 = c2 for a triangle.
Write the converse statement. If
a2 + b2 = c2 for a triangle.
Then
a given triangle is right angled
This converse is true and allows us to find right angled triangles.

35. Testing For A Right Angled Triangle.
Is the triangle below right angled ?
(1) Which side is the longest side ? 6m 8m
10m
10m
(2) Add the sum of the squares of the two shorter sides.
102
(3) Square the longest side separately. =
100
=100
(4) Are the two calculations equal to each other?
As = 10 2 then by the converse of Pythagoras the triangle is right angled.
yes

36. Is the triangle below right angled ?
6.9
9.2
11.5
(1) Which side is the longest side ?
11.5
(2) Add the sum of the squares of the two shorter sides.
11.5 2
(3) Square the longest side separately. =
132.25
=132.25
(4) Are the two calculations equal to each other?
As = then by the converse of Pythagoras the triangle is right angled.
yes

37. Is the triangle below right angled ?
8.1
10.8
14.5
(1) Which side is the longest side ?
14.5
(2) Add the sum of the squares of the two shorter sides.
14.5 2
(3) Square the longest side separately. =
210.25
=182.25
(4) Are the two calculations equal to each other?
As  then by the converse of Pythagoras the triangle is not right angled.
No.

38. Use the converse of Pythagoras to determine if these triangles are right angled or not.
(2) 39
28.8
23.4
(1) 13 5 12
Yes No
(3)
107.9
41.5
99.6
(4) 49
117.6
123.5
Yes No