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

Squaring a Squaring a NumberNumber

To square a number means to :

“Multiply it by itself”

means 9 x 9 = 81

Example :

29

means 10 x 10 = 100210

Squaring a Squaring a NumberNumber

Calculate:

a) 2² b) 4² c) 8² d) 1²

Find: a) 3² + 6² b) 4² + 1² c) 8² + 6² d) 9² + 5²

2 X 2 = 4 4 X 4 = 16 8 X 8 = 16 1 X 1 = 1

3 X 3 + 6 X 6 9 X 9 + 5 X 58 X 8 + 6 X 64 X 4 + 1 X 1

9 + 36 = 45 16 + 1 = 17 64 + 36 = 100 81 + 25 = 106

Square Root of a Square Root of a numbernumber

You now know how to find :

We can ‘undo’ this by asking

““which number, times itself, gives 81”which number, times itself, gives 81”

92 = 9 x 9 = 81

From the top line, the answer is 9

This is expressed as : “the SQUARE ROOT of 81 is 9”“the SQUARE ROOT of 81 is 9”

or in symbols we write : 81=9

Square Root of a Square Root of a numbernumber

Find:

a) √36 b)√25 c) √1 d) √144

A = 49cm²This square has an area of 49cm².What is the length of one of the sides?

A = 4cm²This square has an area of 4cm².What is the length of one of the sides?

= 6 = 1 = 12= 5

√49 = 7

√4 = 2

Right – Angle Right – Angle TrianglesTriangles

In a right angled triangle the side directly across from the right angle is called

The Hypotenu

se

Right – Angle Right – Angle TrianglesTriangles

c

b

a

Measure the length of a ?

Measure the length of b ?

Complete the triangle and measure the length of c (the hypotenuse)

34

5

Right – Angle TrianglesRight – Angle Triangles

c

b

a

Measure the length of a ?Measure the length of b ?

Complete the triangle and measure the length of

C (the hypotenuse)

68

10

Right – Angle Right – Angle TrianglesTriangles

c b

a

Measure the length of a ?Measure the length of b ?

Complete the triangle and measure the length of

c the hypotenuse

5

12

13

Right – Angle TrianglesRight – Angle Triangles

2 2 2a b c

a b c a2 b2 c2

3 4 5 9 16 25

5 12 13 25 144 169

6 8 10 36 64 100

Can anyone spot a

relationship between a2, b2, c2.

c

b

a

Pythagoras’s Pythagoras’s TheoremTheorem

a

bc

2 2 2a b c

Summary of Summary of Pythagoras’s TheoremPythagoras’s Theorem

Note: The equation is ONLY valid for right-

angled triangles.

2 2 2a b c

Pythagoras’s Pythagoras’s TheoremTheorem

Calculate the lengths of the hypotenuse in each case

12cm

9cm

c cm c cm

8cm

15cm

12cm

16cmc cm

Calculating the HypotenuseCalculating the Hypotenuse

8

12

c

Q2. Calculate the longest length of the right-angled triangle below.

2 2 2c =a +b2 2 2c =12 +8

2c =208

c = 208 =14.42km

Example 1

Calculating the Hypotenuse

Aeroplaneb = 8

a = 15

c

Lennoxtown

Airport

Q1.Q1. An aeroplane is preparing to land at Glasgow An aeroplane is preparing to land at Glasgow Airport. Airport. It is over Lennoxtown at present which is It is over Lennoxtown at present which is 15km from 15km from the airport. It is at a height of 8km. the airport. It is at a height of 8km.

How far away is the plane from the airport?How far away is the plane from the airport?

2 2 2c =a +b2 2 2c =15 +8

2c =289

c = 289 =17km

Example 2

Calculating the HypotenuseCalculating the Hypotenuse

Example 1:

Calculate the length of the missing side of this triangle.

8cm

6cm

The Hypotenuse (longest side)

a² + b² = c²

8² + 6² = c²

64 + 36 = c²

c² = 100

c = √100 = 10

Calculating the HypotenuseCalculating the Hypotenuse

Example 2:

Calculate the length of the missing side of this triangle.

12cm

5cm

The Hypotenuse

(longest side)

a² + b² = c²

12² + 5² = c²

144 + 25 = c²

c² = 169

c = √169 = 13

Solving Real-Life ProblemsSolving Real-Life Problems

Example : A steel rod is used to support a treewhich is in danger of falling down.

What is the length of the rod?

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.

2 2 2c =a +b2 2 2c =8 +152c =289

c = 289 =17m

15m

8m

rod

Solving Real-Life ProblemsSolving Real-Life Problems

Example 2A garden is rectangular in shape. A fence is

to be put along the diagonal as shown below. What is the length of the fence.

2 2 2c =a +b2 2 2c =10 +152c =325

c = 325 =18.03m

10m

15m

Length of the smaller sideLength 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.

Pythagoras’ Theorem is : a² + b² = c²

What if we want to find out a value for a?

Length of the smaller sideLength of the smaller side

a² + b² = c²-b² -b²

a² = c² - b²

What side of the triangle is “c”

The Hypotenuse or Longest side

What we do to one side we have to do to

the other

Always ‘take away’ the shorter side from the

longest side

Pythagoras’ Theorem is : a² + b² = c²

What if we want to find out a value for b?

Length of the smaller sideLength of the smaller side

a² + b² = c²-a² -a²

b² = c² - a²

What side of the triangle is “c”

The Hypotenuse or Longest side

What we do to one side we have to do to

the other

Always ‘take away’ the shorter side from the

longest side

Length of the smaller sideLength of the smaller side

Example : Find the length of side a ?

2 2 2c =a +b

2 2 2a =20 -122a =256

a= 256 =16cm 20cm 12cm

a cm

2 2 2a =c - b

Check answer ! Always smaller

than hypotenus

e

Always take small side away

from hypotenuse

when finding a sorter side!!!

Length of the smaller sideLength of the smaller side

Example : Find the length of side b ?

2 2 2c =a +b

2 2 2b =10 - 82b =36

b = 36 =6cm

10cm b cm

8 cm

2 2 2b =c - a

Check answer ! Always smaller

than hypotenus

e

Always take small side away

from hypotenuse

when finding a sorter side!!!

Pythagoras’ Theorem is : a² + b² = c²

What if we want to find out a value for a?

Length of the smaller sideLength of the smaller side

a² + b² = c²

a² = c² - b²

Always ‘take away’ the shorter side from the

longest side

We need to re-arrange Pythagoras’ Theorem to form

Length of the smaller sideLength of the smaller side

Example : Find the length of side a ?

13cm 12cm

a cm

a² = c² - b²

a² = 13² - 12²

a² = 169 - 144

a² = 25

a = √25 = 5

Length of the smaller sideLength of the smaller side

Example : Find the length of side b ?

15cm b cm

12 cm

b² = c² - a²

b² = 15² - 12²

b² = 225 - 144

b² = 81

a = √81 = 9

Pythagoras TheoremPythagoras Theorem

Finding hypotenuse c2 2 2c =a +b

2 2 2a =c - b

2 2 2b =c - a

Finding shorter side a

Findingshorter side b

a

c b