MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Pythagoras Pythagoras TheoremTheorem
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Investigating Pythagoras Theorem
Finding the length of the smaller side
Calculating the Hypotenuse
Solving real-life problems
Distance between two points & Mixed Problems
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Starter Questionsw
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2 2
.
3 4
I s 14 2 the same as Explain?40 5
1. I s (- 2) 3 3 = 3. Explain your answer
2. Calculate 45% of 360
3.
4. Calculate
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Squaring a Squaring a NumberNumber
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1.1. To understand what is To understand what is meant by the termmeant by the term
‘‘squaring a number’squaring a number’
1. We are learning the term ‘‘squaring a number’.squaring a number’.
2.2. Be able to calculate Be able to calculate squares both mentally and squares both mentally and using the calculator.using the calculator.
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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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
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Exercise 1Ch59 (page 202)
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Squaring a Squaring a NumberNumber
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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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
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Exercise 2Ch59 (page 203)
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Square Root of a Square Root of a NumberNumber
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
2 2 = 41 where a = 4 and b = (-5). Explain?I s a b
Starter QuestionsStarter Questions
Q1.Q1. Are the missing anglesAre the missing angles6565oo, 40, 40oo and 65 and 65oo
Q2.Q2. CalculateCalculate
Q3. Q3. The cost of a new computer is £1000 +vat. If the vat is The cost of a new computer is £1000 +vat. If the vat is charged at 12% what is the total cost.charged at 12% what is the total cost.
Q4. Q4. The cost of a bag of sugar is £1.12. How much 50 bags The cost of a bag of sugar is £1.12. How much 50 bags cost.cost.
115o
NON-CALCULATORNON-CALCULATOR
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MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
‘‘To investigate the right-angle triangle and to come To investigate the right-angle triangle and to come up with a relationship between the lengths of its two up with a relationship between the lengths of its two shorter sides and the longest side which is called the shorter sides and the longest side which is called the hypotenuse. ‘hypotenuse. ‘
c
b
a
Aim of today's LessonAim of today's Lesson
Right – Angle Right – Angle TrianglesTriangles
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MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Right – Angle Right – Angle TrianglesTriangles
c
b
a
What is the length of a ?What is the length of b ?
Copy the triangle into your jotter and measure the length of c
34
5
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MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Right – Angle TrianglesRight – Angle Triangles
c
b
a
What is the length of a ?What is the length of b ?
Copy the triangle into your jotter and measure the length of c
68
10
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19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Right – Angle Right – Angle TrianglesTriangles
c b
a
What is the length of a ?What is the length of b ?
Copy the triangle into your jotter and measure the length of c
5
12
13
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MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Right – Angle Right – Angle TrianglesTriangles
a b c a2 b2 c2
3 4 5
5 12 13
6 8 10
c
b
a
Copy the table below and fill in the values that are missing
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MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
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
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MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Pythagoras’s Pythagoras’s TheoremTheorem
a
bc
2 2 2a b c
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MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Summary of Summary of Pythagoras’s TheoremPythagoras’s Theorem
Note: The equation is ONLY valid for right-angled triangles.
2 2 2a b c
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MTH 4-16a
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Exercise 359(page 204)
Pythagoras Theorem
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MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Starter Questionsw
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1. I s 4 - 3 4 = 4. Explain
2. Write 88% as a decimal
3. Why is the LCM of 12 and 7 equal to 84.
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Calculating Calculating HypotenuseHypotenuse
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1. Remember the term hypotenuse “ the longest side”
1. We are learning to use Pythagoras Theorem to calculate the length of the hypotenuse
“the longest side”2. Apply Pythagoras Theorem
to calculate the hypotenuse.
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Calculating the HypotenuseCalculating the Hypotenuse
8
12
c
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
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Example 1MTH 4-16a
19 Apr 2023Compiled by Mr. Lafferty Maths
Dept.
Calculating the Hypotenuse
Aeroplaneb = 8
a = 15
c
Lennoxtown
Airport
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
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Example 2MTH 4-16a
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Exercise 4Ch59 (page 205)
Calculating Calculating HypotenuseHypotenuse
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MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Starter Starter QuestionsQuestions
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24 , 3 2x y and z z xy
1. Calculate 0.5% of £45
2. I f . Find
3. Calculate 60% of £360
4. Solve 2x+3 = x + 11
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Solving Real-Life Solving Real-Life ProblemsProblems
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1.1. Be able to solve real-life Be able to solve real-life problems using problems using Pythagoras Theorem Pythagoras Theorem showing clear working.showing clear working.
1. We are learning how Pythagoras Theorem can be used to solve real-life problems.
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Solving Real-Life Solving Real-Life ProblemsProblems
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
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15m
8m
rod
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Solving Real-Life Solving Real-Life ProblemsProblems
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
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10m
15m
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Exercise 5Ch59 (page 208)
Solving Real-Life Solving Real-Life ProblemsProblems
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MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Starter Questionsw
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5 , 2 4
27
(2 4 )
x
x y and z y z x
1. Solve
2. I f . Find
3. Calculate 80% of £700
4. Find the median, mode and mean for
6, 7, 6, 4, 3, 10
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Length of the smaller Length of the smaller sideside
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1.1. Apply Pythagoras Apply Pythagoras Theorem to find the Theorem to find the length of smaller side length of smaller side showing clear working.showing clear working.
1. We are learning how Pythagoras Theorem can be used to find the length of the smaller side.
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Length of the smaller Length of the smaller sideside
Example : Find the length of side a ?
To find the length of the smaller side of a right- angled
triangle we simply rearrange Pythagoras Theorem.
2 2 2c =a +b
2 2 2a =20 -122a =256
a= 256 =16cm
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20cm 12cm
a cm
2 2 2a =c - bCheck answer ! Always smaller
than hypotenus
e
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Length of the smaller Length of the smaller sideside
Example : Find the length of side b ?
2 2 2c =a +b
2 2 2b =10 - 82b =36
b = 36 =6cm
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10cm b cm
8 cm
2 2 2b =c - a
Check answer ! Always smaller
than hypotenus
e
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Exercise 6Ch59 (page 211)
Length of smaller Length of smaller sideside
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MTH 4-16a
Starter Questionsw
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ALWAYS comes up in
exam !!
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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1.1. Apply Pythagoras Theorem Apply Pythagoras Theorem to find length of a line.to find length of a line.
1. We are learning how Pythagoras Theorem can be used to find the length of a line.
2.2. Show all working.Show all working.
Finding the Finding the Length of a LineLength of a Line
MTH 4-16a
19 Apr 2023Created by Mr. Lafferty Maths
Dept.w
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1 32 540
1
3
2
5
4
1 32 40
1
32
54
6
5 76 8 9 10
78
(7,7)
(2,4)
3
5
Finding the Length of a Line
Discuss with your partner how we might find the length of the
line.
2 2 2c a b
2 2 25 3c
25 9c
34c
5.83c
19 Apr 2023Created by Mr. Lafferty Maths
Dept.w
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1 32 540
1
3
2
5
4
1 32 40
1
32
54
6
5 76 8 9 10
78
(9,1)
(0,6)
5
9
2 2 2c a b
2 2 25 9c
25 81c
106c
10.3c
Pythagoras Theorem to find the length of a LineMTH 4-16a
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Pythagoras Pythagoras TheoremTheorem
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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
A
B
C D E F
G H
IJ
K L
P(x,y)
o
r
M
N
O
P
Q R
S
T
U V
W Z
AC
MTH 4-16a
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Exercise 7Ch59 (page 213)
Pythagoras Pythagoras TheoremTheorem
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