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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