1. Squaring a Number To square a number means to :
“Multiply it by itself”
Example :
means 9 x 9 = 81
means 10 x 10 = 100

2. Squaring a Number Calculate: a) 2² b) 4² c) 8² d) 1² Find:
2 X 2 = 4
4 X 4 = 16
8 X 8 = 16
1 X 1 = 1
Find:
a) 3² + 6² b) 4² + 1² c) 8² + 6² d) 9² + 5²
3 X X 6
4 X X 1
8 X X 6
9 X X 5
= 45
= 17
= 100
= 106

3. Square Root of a number You now know how to find : 92 = 9 x 9 = 81
We can ‘undo’ this by asking
“which number, times itself, gives 81”
From the top line, the answer is 9
This is expressed as : “the SQUARE ROOT of 81 is 9”
or in symbols we write :

4. Square Root of a number Find: a) √36 b)√25 c) √1 d) √144 = 12 = 6 = 5
A = 49cm²
This square has an area of 49cm².
What is the length of one of the sides?
√49 = 7
A = 4cm²
This square has an area of 4cm².
What is the length of one of the sides?
√4 = 2

5. Right – Angle Triangles
In a right angled triangle the side directly
across from the right angle is called
The Hypotenuse

6. Right – Angle Triangles
Measure the length of a ? 3 4
Measure the length of b ?
Complete the triangle
and measure the length
of c (the hypotenuse) 5

7. Right – Angle Triangles
Measure the length of a ? 6
Measure the length of b ? 8
Complete the triangle
and measure the length of
C (the hypotenuse) 10

8. Right – Angle Triangles
Measure the length of a ? 5
Measure the length of b ? 12
Complete the triangle
and measure the length of
c the hypotenuse 13

9. Right – Angle Triangles
Can anyone spot a relationship between
a2, b2, c2. a b c a2 b2 c2 3 4 5 9 16 25 12 13
144
169 6 8 10 36 64
100

10. Pythagoras’s Theorem c b a

11. Summary of Pythagoras’s Theorem
Note: The equation is ONLY valid for right-angled triangles.

12. Pythagoras’s Theorem
Calculate the lengths of the hypotenuse in each case
c cm
c cm
12cm
15cm
9cm
8cm
12cm
16cm
c cm

13. Calculating the Hypotenuse
Example 1
Q2. Calculate the longest length of the right- angled triangle below. c 8 12

14. Calculating the Hypotenuse
Example 2
Q1. An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present which is 15km from the airport. It is at a height of 8km.
How far away is the plane from the airport?
Aeroplane c
b = 8
Airport
a = 15
Lennoxtown

15. Calculating the Hypotenuse
Example 1:
Calculate the length of the missing side of this triangle.
The Hypotenuse (longest side)
6cm
a² + b² = c²
8² + 6² = c²
= c²
c² = 100
c = √100 = 10
8cm

16. Calculating the Hypotenuse
Example 2:
Calculate the length of the missing side of this triangle.
5cm
The Hypotenuse (longest side)
a² + b² = c²
12² + 5² = c²
= c²
c² = 169
c = √169 = 13
12cm

17. Solving Real-Life Problems
When coming across a problem involving finding a missing side in a right-angled triangle, you should consider using Pythagoras’ Theorem to calculate its length.
Example : A steel rod is used to support a tree
which is in danger of falling down.
What is the length of the rod?
15m 8m
rod

18. Solving Real-Life Problems
Example 2
A garden is rectangular in shape. A fence is to be put along the diagonal as shown below.
What is the length of the fence.
10m
15m

19. Length of the smaller side
To find the formula for calculating
the length of a
smaller side we have to re-arrange
Pythagoras’
using the balancing method we learned in our
Brackets and equation topic.

20. Length of the smaller side
The Hypotenuse or Longest side
What side of the triangle is “c”
Length of the smaller side
Pythagoras’ Theorem is :
a² + b² = c²
What if we want to find out a value for a?
What we do to one side we have to do to the other
a² + b² = c²
-b²
-b²
a² = c² - b²
Always ‘take away’ the shorter side from the longest side

21. Length of the smaller side
The Hypotenuse or Longest side
What side of the triangle is “c”
Length of the smaller side
Pythagoras’ Theorem is :
a² + b² = c²
What if we want to find out a value for b?
What we do to one side we have to do to the other
a² + b² = c²
-a²
-a²
b² = c² - a²
Always ‘take away’ the shorter side from the longest side

22. Length of the smaller side
Always take small side away from hypotenuse when finding a sorter side!!!
Example : Find the length of side a ?
Check answer ! Always smaller than hypotenuse
20cm
12cm
a cm

23. Length of the smaller side
Always take small side away from hypotenuse when finding a sorter side!!!
Example : Find the length of side b ?
10cm
b cm
8 cm
Check answer ! Always smaller than hypotenuse

24. Length of the smaller side
Pythagoras’ Theorem is :
a² + b² = c²
What if we want to find out a value for a?
a² + b² = c²
We need to re-arrange Pythagoras’ Theorem to form
a² = c² - b²
Always ‘take away’ the shorter side from the longest side

25. Length of the smaller side
Example : Find the length of side a ?
a² = c² - b²
a² = 13² - 12²
a² =
a² = 25
a = √25 = 5
13cm
12cm
a cm

26. Length of the smaller side
Example : Find the length of side b ?
b² = c² - a²
b² = 15² - 12²
b² =
b² = 81
a = √81 = 9
15cm
b cm
12 cm

27. Pythagoras Theorem Finding hypotenuse c Finding shorter side b c b a
Finding shorter side a