1-aug-15compiled by mr. lafferty maths dept. pythagoras theorem investigating pythagoras theorem...
TRANSCRIPT
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 the length of the smaller side
Solving Real – Life Problems
Pythagoras Theorem Twice
Converse of Pythagoras Theorem
S3 Credit
19 Apr 2023 Compiled by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questions
Q1.Q1. Explain why 4 + 6 Explain why 4 + 6 xx 5 = 34 and not 50 5 = 34 and not 50
Q2.Q2. CalculateCalculate
2 2 where a = 4 and b = (-5)a b
Q3. Q3. Does xDoes x22 – 121 factorise to (x – 11) (x - 11) – 121 factorise to (x – 11) (x - 11)
Q4. Q4. The cost of an iPod is £80 including VAT. The cost of an iPod is £80 including VAT. How much is the iPod BEFORE VAT.How much is the iPod BEFORE VAT.
NON-CALCULATORNON-CALCULATOR
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‘‘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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19 Apr 2023 Compiled 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
3
4
5
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19 Apr 2023 Compiled 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
6
8
10
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19 Apr 2023 Compiled 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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19 Apr 2023 Compiled 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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19 Apr 2023 Compiled 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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19 Apr 2023 Compiled by Mr. Lafferty Maths Dept.
Pythagoras’s Pythagoras’s TheoremTheorem
a
bc
2 2 2a b c
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19 Apr 2023 Compiled by Mr. Lafferty Maths Dept.
Pythagoras’s Pythagoras’s TheoremTheorem
y
x
z
2 2 2( ) ( ) ( )xy yz xz
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19 Apr 2023 Compiled 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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19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Calculating Calculating HypotenuseHypotenuse
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1.1. Know the term Know the term hypotenuse hypotenuse “ the “ the longest side”longest side”
1. Use Pythagoras Theorem to calculate the length of the hypotenuse
““the longest side”the longest side” 2.2. Use Pythagoras Theorem Use Pythagoras Theorem to calculate the to calculate the hypotenuse.hypotenuse.
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Calculating Calculating HypotenuseHypotenuse
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Two key points when dealing with right-angled triangles
The longest side in a right-angled triangle is calledThe HYPOTENUSE
The HYPOTENUSE is ALWAYS opposite the right angle
a
b cc2 = a2 + b2
y
x
z
(xz)2 = (xy)2 + (yz)2
S3 Credit
19 Apr 2023 Compiled by Mr. Lafferty Maths Dept.
Calculating the HypotenuseCalculating the Hypotenuse
8
12
c
Q2.Q2. Calculate the longest length of the right-Calculate the longest length of the right-angled triangle below.angled triangle below.
2 2 2c =a +b2 2 2c =12 +8
2c =208
c = 208 =14.42km
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Example 1Example 1
19 Apr 2023 Compiled by Mr. Lafferty Maths Dept.
Calculating the HypotenuseCalculating the Hypotenuse
LA = 8
GL = 15
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?
( )GA 2 2 2=(GL) +(LA)
( )GA 2 2 2=15 +8
( )GA 2 =289
GA = 289 =17km
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Example 2Example 2
A
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19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Ex 2.1 and 2.2MIA Page 147
Calculating Calculating HypotenuseHypotenuse
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S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
S3 Starter QuestionsS3 Starter Questionsw
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2 2
5
(2 3 4)
8 4 (2
4 45 115 9
4 1)
3 65 25
1. Does
2. Expand
3. Does factorise to
4.
y y x
d d dd
S3 Credit
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. Use Pythagoras Use Pythagoras Theorem to find the Theorem to find the length of smaller side.length of smaller side.
1. To show how Pythagoras Theorem can be used to find the length of the smaller side.
2.2. Show all working.Show all working.
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Example : Find the length of side a ?
Length of the smaller Length of the smaller sideside
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
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Example : Find the length of side a ?
Length of the smaller Length of the smaller sideside
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
S3 Credit
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
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now work throughEx3.1 and Ex 3.2
Odd Numbers Only
Length of smaller Length of smaller sideside
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S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
S3 Starter QuestionsS3 Starter Questionsw
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-4
2
45 2.56 10
65% £40 £26
3 1 16 14 4 16
- 6 8
1. Calculate
2. Write in scientific notation
3. Factorise
4. Explain why
k
of
k
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Solving Real – Life Solving Real – Life ProblemsProblems
Using Pythagoras Using Pythagoras TheoremTheorem
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1.1. Apply Pythagoras Theorem Apply Pythagoras Theorem to solve real-life problems.to solve real-life problems.
1. To show how Pythagoras Theorem can be used to solve real-life problems.
2.2. Show all working.Show all working.
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Example : A steel rod is used to support a treewhich is in danger of falling down.
What is the height of the tree ?
Solving Real-Life Solving Real-Life ProblemsProblems
When coming across a problem involving When coming across a problem involving finding a missing side in a right-angled triangle, finding a missing side in a right-angled triangle, you should consider using Pythagoras’ Theorem you should consider using Pythagoras’ Theorem to calculate its length.to calculate its length.
2 2 2c a b
2 2 217 - 8a
2 225a
225 15a m
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17m
8m
rod
b
ac
S3 Credit
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 2Example 2A garden has a fence around its perimeter and A garden has a fence around its perimeter and along its diagonal as shown below. along its diagonal as shown below. What is the length of the fence from D to C.What is the length of the fence from D to C.
2 2 2( ) ( ) ( )AC DC AD
2 2 2( ) 13 - 5DC
2( ) 144DC
144 12DC m
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5m
b m
13m
A B
CD
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now work throughEx4.1 and Ex 4.2
Odd Numbers Only
Length of smaller Length of smaller sideside
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S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
S3 Starter QuestionsS3 Starter Questionsw
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2 2
2
4 , 3 2
24 16
3 1 15 2 74 8 8
2
1. True or false
2. I f .
Find
3. Factorise
p q and r
q
y y
z r
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Pythagoras Pythagoras TheoremTheorem
TwiceTwice
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1.1. Use the appropriate Use the appropriate form of Pythagoras form of Pythagoras Theorem to solving Theorem to solving harder problems.harder problems.
1. To use knowledge already gained on Pythagoras Theorem to solve harder problems using Theorem twice. 2.2. Show all working.Show all working.
S3 Credit
19 Apr 202319 Apr 2023
Solving Real-Life Solving Real-Life ProblemsProblems
Problem : Find the length of h.
2 2 2( ) ( ) ( )BD AD AB
2 2 2( ) 15 13BD
2( ) 394BD
394BD
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h
A
B C
D
12
13
15
Find length BD first
S3 Credit
19.85BD
19.85
19 Apr 202319 Apr 2023
Solving Real-Life Solving Real-Life ProblemsProblems
Problem : Find the length of length y.
2 2 2( ) ( ) ( )BC BD DC
2 2 2( ) ( )h BD DC
2 2 2(19.85) (12)h
2 394 144h
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h
A
B C
D
12
13
15
Now find h
538h
23.2 ( 1 . .)h to d p
S3 Credit
394
19 Apr 202319 Apr 2023
Solving Real-Life Solving Real-Life ProblemsProblems
Problem : Find the diagonal length of the cuboid AG.
2 2 2( ) ( ) ( )AH AD DH
2 2 2( ) 8 6AH
2( ) 64 36AH
100AH
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Find AH first
10AH cm A
B C
D
E
F G
H6cm
8cm
7cm
10cm
S3 Credit
19 Apr 202319 Apr 2023
Solving Real-Life Solving Real-Life ProblemsProblems
Problem : Find the diagonal length of the cuboid AG.
2 2 2( ) ( ) ( )AG AH HG
2 2 2( ) 10 7AG
2( ) 100 49AG
2( ) 149AG
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Now find AG
12.2 ( 1 . .)AG cm to d p
S3 Credit
A
B C
D
E
F G
H6cm
8cm
7cm
10cm
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Ex 5.1Ch8 (page 154)
Pythagoras Pythagoras TheoremTheorem
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S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
S3 Starter QuestionsS3 Starter Questionsw
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2 2 2 2
(1 )(2 3 )
1
1 1 16 7 42
- ( )
2 515 3
1. True or false
2. Expand
3. Does factorise to
4.
w w
ab ba b a a b
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Converse of Converse of Pythagoras Pythagoras
TheoremTheorem
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1.1. Apply the converse of Apply the converse of Pythagoras Theorem to Pythagoras Theorem to prove a triangle is prove a triangle is right-angled.right-angled.
1. To explain the converse of Pythagoras Theorem to prove a triangle is right-angled.
S3 Credit
2. Right-angle is directly opposite C.
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Converse of Converse of Pythagoras Pythagoras
TheoremTheorem
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2 2 2a +b =cb
a
Converse Theorem states that if
S3 Credit
1. Then triangle MUST be right-angled.
c
Hypotenuse
Converse1 – talk
Converse2 – opposite, reverse
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Converse of Converse of Pythagoras Pythagoras
TheoremTheorem
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Problem : Is this triangle right-angled ? Explain Answer
2 2 2a +b =c
2c =100
117 100
10cm
6 cm
9 cm
2 2 2 2a +b =9 +6 =81+36 = 117
If it is then Pythagoras Theorem will be true
By the Converse Theorem, triangle is NOT right-angled
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Converse of Converse of Pythagoras Pythagoras
TheoremTheorem
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Problem : A picture frame manufacturer claims that his are rectangular is his claim true.
2 2 2a +b =c
2c =25002 2 2a +b =c 30 cm
40 cm2 2 2 2a +b =40 +30 =1600 +900 = 2500
If it is then Pythagoras Theorem will be true
By the Converse Theorem, frame IS rectangular
50cm
S3 Credit
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try Ex 6.1 & 6.2
Ch8 (page 156)
Converse of Converse of Pythagoras Pythagoras
TheoremTheorem
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S3 Credit