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Pythagoras Pythagoras TheoremTheorem
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Revision of simple Pythagoras Theorem
Converse of Pythagoras Theorem
3D - Pythagoras Theorem
Exam Questions
Nat 5
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questionsw
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Given the two straight lines
3y + 2x - 12 = 0 4x + 6y = -24
In pairs explain what they have in commonAnd where each cross the x-axis and y-axis.
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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1.1. Know the term Know the term hypotenuse hypotenuse “ the “ the longest side”longest side”
1. We are revising Pythagoras Theorem
2.2. Use Pythagoras Theorem Use Pythagoras Theorem to calculate both the to calculate both the hypotenuse and smaller hypotenuse and smaller sides of a right-angled sides of a right-angled triangle.triangle.
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Revision of Pythagoras Revision of Pythagoras TheoremTheorem
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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
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Revision of Pythagoras Revision of Pythagoras TheoremTheorem
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 ?
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
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Revision of Pythagoras Revision of Pythagoras TheoremTheorem
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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
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Revision of Pythagoras Revision of Pythagoras TheoremTheorem
MTH 4-16a
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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
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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
19 Apr 202319 Apr 2023
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
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19.85BD
19.85
Revision of Pythagoras Revision of Pythagoras TheoremTheorem
19 Apr 202319 Apr 2023
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
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394
Revision of Pythagoras Revision of Pythagoras TheoremTheorem
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Revision of Pythagoras Revision of Pythagoras TheoremTheorem
NOTE !!!!
Pythagoras Theorem tends to pop up within circle type question
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 try N5 TJEx5.1
Ch5 (page 44)
Revision of Pythagoras Revision of Pythagoras TheoremTheorem
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19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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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
Nat 5
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. We are learning the converse of Pythagoras Theorem to prove a triangle is right-angled.
Nat 5
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
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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
2c =10210cm
6 cm
9 cm
2 2a +b
If it is then Pythagoras Theorem will be true.
Nat 5
2 2= 9 +6=81+36 = 117
2c =100
Since 117 ≠ 100 then by the Converse Theorem of
Pythagoras triangle IS NOT right-angled.
a
b
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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Problem : A picture frame manufacturer claims that his are rectangular is his claim true.
30 cm
40 cm
If it is then Pythagoras Theorem will be true
50cm
Nat 5
2c =5022 2a +b2 2= 30 +40
=900 +1600 = 2500
2c =2500
Since 2500 = 2500 then by the Converse Theorem of Pythagoras triangle IS
right-angled.
a
b
c
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N5 TJEx5.2
Ch5 (page 46)
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Converse of Converse of Pythagoras Pythagoras
TheoremTheorem
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
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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
Nat 5
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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1.1. Use the appropriate Use the appropriate form of Pythagoras form of Pythagoras Theorem to solving 3D Theorem to solving 3D problems.problems.
1. We are learning how to apply knowledge already gained on Pythagoras Theorem to solve 3D problems.
2.2. Show all working.Show all working.
Nat 5
3D Pythagoras 3D Pythagoras TheoremTheorem
19 Apr 202319 Apr 2023
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
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3D Pythagoras 3D Pythagoras TheoremTheorem
19 Apr 202319 Apr 2023
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
Nat 5
A
B C
D
E
F G
H6cm
8cm
7cm
10cm
3D Pythagoras 3D Pythagoras TheoremTheorem
19 Apr 202319 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N5 TJEx5.3
Ch5 (page 47)
3D Pythagoras 3D Pythagoras TheoremTheorem
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