Transcript Pythagoras Theorem [PPS]

Pythagoras Theorem
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S3
Credit
Investigating Pythagoras Theorem
Finding the length of the smaller side
Solving Real – Life Problems
Pythagoras Theorem Twice
Converse of Pythagoras Theorem
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Starter Questions
Q1.
Explain why 4 + 6
Q2.
Calculate
a 2  b2
x
5 = 34 and not 50
where a = 4 and b = (-5)
Q3.
Does x2 – 121 factorise to (x – 11) (x - 11)
Q4.
The cost of an iPod is £80 including VAT.
How much is the iPod BEFORE VAT.
NON-CALCULATOR
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Right – Angle Triangles
c
b
Aim of today's Lesson
‘To investigate the right-angle triangle and to come up
with a relationship between the lengths of its two shorter
sides and the longest side which is called the hypotenuse. ‘
a
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Right – Angle Triangles
What is the length of
c
b
What is the length of b ?
4
Copy the triangle into your jotter
and measure the length of c
5
a
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a? 3
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Right – Angle Triangles
What is the length of
a?6
What is the length of b ?
8
c
b
Copy the triangle into your jotter
and measure the length of c
10
a
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Right – Angle Triangles
What is the length of
a? 5
What is the length of b ? 12
c
b
Copy the triangle into your jotter
and measure the length of c
13
a
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Right – Angle Triangles
Copy the table below and fill in the values that are missing
a
b
c
3
4
5
a2
b2
c2
c
b
5
12
13
6
8
10
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a
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Right – Angle Triangles
a
b
c
a2
b2
c2
3
4
5
9
16
25
5
12
13
25
144
169
6
8
10
36
64
100
a2  b2  c 2
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Can anyone
spot a
relationship
between
a2, b2, c2.
c
b
a
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Pythagoras’s Theorem
c
b
a b c
2
a
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2
2
Pythagoras’s Theorem
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x
y
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( xy )  ( yz)  ( xz)
2
2
z
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2
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Summary of
Pythagoras’s Theorem
a b c
2
2
2
Note: The equation is ONLY valid
for right-angled triangles.
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Calculating Hypotenuse
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Credit
Learning Intention
1. Use Pythagoras Theorem
to calculate the length of
the hypotenuse
“the longest side”
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Success Criteria
1. Know the term hypotenuse
“ the longest side”
2. Use Pythagoras Theorem to
calculate the hypotenuse.
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Calculating Hypotenuse
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Two key points when dealing with
right-angled triangles
The longest side in a right-angled triangle is called
The HYPOTENUSE
The HYPOTENUSE is ALWAYS opposite the right angle
c
b
c 2 = a 2 + b2
(xz)2 = (xy)2 + (yz)2
x
a
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y
z
Calculating the Hypotenuse
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Example 1
Q2.
Calculate the longest length of the rightangled triangle below.
c
c2 = a2 + b2
8
c2 = 122 + 82
c2 = 208
12
c = 208 =14.42km
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Calculating the Hypotenuse
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Example 2
Q1. An aeroplane is preparing to land at Glasgow Airport.
It is over Lennoxtown at present which is 15km from
the airport. It is at a height of 8km.
How far away is the plane from the airport?
(GA)2 = (GL)2 + (LA)2
A
(GA)2 =152 + 82
LA = 8
(GA) = 289
2
GA = 289 =17km
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G
L
Airport GL = 15 Lennoxtown
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Calculating Hypotenuse
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S3
Credit
Now try
Ex 2.1 and 2.2
MIA Page 147
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S3 Starter Questions
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4
4
5  5 11
5
9
1.
Does
2.
Expand
3.
Does 8d 2  4d factorise to 4d 2 (2d 1)
4.
3 6

5 25
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y(2 y  3x  4)
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Length of the smaller side
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Learning Intention
1. To show how Pythagoras
Theorem can be used to find
the length of the smaller side.
Success Criteria
1. Use Pythagoras Theorem
to find the length of
smaller side.
2. Show all working.
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Length of the smaller side
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To find the length of the smaller side
of a right-angled triangle
we simply rearrange Pythagoras Theorem.
Example :
Check
answer !
Always
smaller than
hypotenuse
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Find the length of side a ?
c2 = a2 + b2
a2 = c2 -b2
a2 = 202 -122
a2 = 256
a = 256 =16cm
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20cm
a cm
12cm
Length of the smaller side
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Example :
Find the length of side a ?
c2 = a2 + b2
b2 = c2 - a2
Check
answer !
Always
smaller than
hypotenuse
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10cm
b2 = 102 - 82
b2 = 36
b = 36 = 6cm
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8 cm
b cm
N
G
H
O
C
D
E
F
M
J
I
K
r
L
C
Q
U
A
R
V
T
A
W
P
Z
S
B
o
P(x,y)
Length of smaller side
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Credit
Now work through
Ex3.1 and Ex 3.2
Odd Numbers Only
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S3 Starter Questions
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3 1 16
  1
4 4 16
1.
Calculate
2.
Write in scientific notation
45  2.56  10-4
3.
Factorise k 2 - 6k  8
4.
Explain why 65% of £40  £26
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Solving Real – Life Problems
Using Pythagoras Theorem
Learning Intention
1. To show how Pythagoras
Theorem can be used to solve
real-life problems.
Success Criteria
1. Apply Pythagoras Theorem
to solve real-life problems.
2. Show all working.
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Solving Real-Life Problems
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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.
Example :
c 2  a2  b2
a  17 - 8
2
2
2
A steel rod is used to support a tree
which is in danger of falling down.
What is the height of the tree ?
a2  225
c
17m
a  225  15m
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rod
8m b
a
Solving Real-Life Problems
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Example 2
A garden has a fence around its perimeter and
along its diagonal as shown below.
What is the length of the fence from D to C.
( AC)2  (DC)2  ( AD)2 A
(DC)2  132 - 52
5m
B
13m
(DC)2  144
DC  144  12m
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D
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bm
C
Length of smaller side
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Now work through
Ex4.1 and Ex 4.2
Odd Numbers Only
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S3 Starter Questions
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S3
Credit
1.
3 1 1
True or false 5  2  7
4 8 8
2.
If p  4 , q  3 and r  2.
Find 2q 2 z  r 2
3.
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Factorise
24 y 16 y 2
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Pythagoras Theorem
Twice
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Learning Intention
1. To use knowledge already
gained on Pythagoras Theorem
to solve harder problems using
Theorem twice.
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Success Criteria
1. Use the appropriate form
of Pythagoras Theorem to
solving harder problems.
2. Show all working.
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Solving Real-Life Problems
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Credit
Problem : Find the length of h.
Find length BD first
(BD)  ( AD)  ( AB)
2
2
2
(BD)  15  13
15
(BD)2  394
A
2
BD  394
BD  19.85
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2
2
h
B
19.85
13
C
12
D
Solving Real-Life Problems
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Credit
Problem : Find the length of length y.
Now find h
(BC)  (BD)  (DC)
2
2
h2  (BD)2  (DC)2
2
h
B
15
394
12
h2  (19.85)2  (12)2
h2  394  144
h  538
h  23.2 (to 1 d.p.)
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A
13
C
D
Solving Real-Life Problems
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Credit
Problem : Find the diagonal length of the cuboid AG.
Find AH first
F
( AH )  ( AD)  (DH )
2
2
( AH )2  82  62
2
B
( AH )2  64  36
AH  100
AH  10cm
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G
C
E
A
8cm
D
7cm
H
6cm
Solving Real-Life Problems
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Credit
Problem : Find the diagonal length of the cuboid AG.
Now find AG
F
( AG)  ( AH )  (HG)
2
2
2
( AG)2  102  72
B
( AG)2  100  49
( AG)2  149
AG  12.2cm ( to 1 d.p.)
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G
C
E
A
8cm
D
7cm
H
6cm
Pythagoras Theorem
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Credit
Now try Ex 5.1
Ch8 (page 154)
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S3 Starter Questions
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S3
Credit
1 1 1
 
6 7 42
1.
True or false
2.
Expand (1  w)(2  3w)
3.
Does ab2 - a2b factorise to b2a2 (a  b)
2 5
4. 1 1
5 3
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Converse of
Pythagoras Theorem
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Learning Intention
1. To explain the converse of
Pythagoras Theorem to prove a
triangle is right-angled.
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Success Criteria
1. Apply the converse of
Pythagoras Theorem to
prove a triangle is
right-angled.
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Converse1 – Converse
talk
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of
2 – opposite,
Pythagoras
Theorem
Converse
reverse
Converse Theorem states that if
a +b = c
2
2
2
c
b
a
1.
Then triangle MUST be right-angled.
2.
Right-angle is directly opposite C.
Hypotenuse
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Converse of
Pythagoras Theorem
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Credit
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Problem : Is this triangle right-angled ? Explain Answer
If it is then Pythagoras Theorem will be true
a2 + b2 = c2
a + b = 9 + 6 = 81 + 36 = 117
2
2
2
2
c = 100
10cm
6 cm
2
117  100
9 cm
By the Converse Theorem, triangle is NOT right-angled
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Converse of
Pythagoras Theorem
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Problem : A picture frame manufacturer claims that his
are rectangular is his claim true.
If it is then Pythagoras Theorem will be true
a2 + b2 = c2
a + b = 40 + 30 = 1600 + 900 = 2500
2
2
2
2
40 cm
50cm
c2 = 2500
a2 + b2 = c2
By the Converse Theorem, frame IS rectangular
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30 cm
Converse of
Pythagoras Theorem
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Credit
Now try Ex 6.1 & 6.2
Ch8 (page 156)
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