Pythagoras Theorem

1
Pythagoras Theorem
S3 Credit
Investigating Pythagoras Theorem
Finding the length of the smaller side
Solving Real Life Problems
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Pythagoras Theorem Twice
Converse of Pythagoras Theorem
2
Starter Questions
Q1. Explain why 4 6 x 5 34 and not 50
Q2. Calculate
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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
3
Right Angle Triangles
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.
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4
Right Angle Triangles
What is the length of a ?
3
4
What is the length of b ?
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Copy the triangle into your jotter and measure
the length of c
5
5
Right Angle Triangles
What is the length of a ?
6
8
What is the length of b ?
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Copy the triangle into your jotter and measure
the length of c
10
6
Right Angle Triangles
What is the length of a ?
5
12
What is the length of b ?
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Copy the triangle into your jotter and measure
the length of c
13
7
Right Angle Triangles
Copy the table below and fill in the values that
are missing
a b c a2 b2 c2
3 4 5
5 12 13
6 8 10
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Right Angle Triangles
Can anyone spot a relationship between a2, b2,
c2.
a b c a2 b2 c2
3 4 5 9 16 25
5 12 13 25 144 169
6 8 10 36 64 100
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Pythagorass Theorem
c
b
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a
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Pythagorass Theorem
x
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y
z
11
Summary of Pythagorass Theorem
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Note The equation is ONLY valid for right-angled
triangles.
12
Calculating Hypotenuse
S3 Credit
Learning Intention
Success Criteria

1. Know the term hypotenuse the longest side

• Use Pythagoras Theorem to calculate the length
  of the hypotenuse
• the longest side

1. Use Pythagoras Theorem to calculate the
   hypotenuse.

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Calculating Hypotenuse
S3 Credit
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
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(xz)2 (xy)2 (yz)2
c2 a2 b2
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Calculating the Hypotenuse
Example 1
Q2. Calculate the longest length of the
right- angled triangle below.
c
8
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12
15
Calculating the Hypotenuse
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?
A
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LA 8
L
G
GL 15
Airport
Lennoxtown
16
Calculating Hypotenuse
S3 Credit
Now try Ex 2.1 and 2.2 MIA Page 147
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S3 Starter Questions
S3 Credit
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18
Length of the smaller side
S3 Credit
Learning Intention
Success Criteria

1. Use Pythagoras Theorem to find the length of
   smaller side.

1. To show how Pythagoras Theorem can be used to
find the length of the smaller side.
2. Show all working.
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Length of the smaller side
S3 Credit
To find the length of the smaller side of a
right-angled triangle we simply rearrange
Pythagoras Theorem.
Example Find the length of side a ?
Check answer ! Always smaller than hypotenuse
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Length of the smaller side
S3 Credit
Example Find the length of side a ?
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Check answer ! Always smaller than hypotenuse
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(No Transcript)
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Length of smaller side
S3 Credit
Now work through Ex3.1 and Ex 3.2 Odd Numbers
Only
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S3 Starter Questions
S3 Credit
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Solving Real Life ProblemsUsing Pythagoras
Theorem
S3 Credit
Learning Intention
Success Criteria

1. Apply Pythagoras Theorem to solve real-life
   problems.

1. To show how Pythagoras Theorem can be used to
solve real-life problems.
2. Show all working.
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Solving Real-Life Problems
S3 Credit
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 A steel rod is used to support a
tree which is in danger of falling
down. What is the height of the tree ?
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a
c
b
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Solving Real-Life Problems
S3 Credit
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.
A
B
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13m
5m
C
D
b m
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Length of smaller side
S3 Credit
Now work through Ex4.1 and Ex 4.2 Odd Numbers
Only
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S3 Starter Questions
S3 Credit
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Pythagoras TheoremTwice
S3 Credit
Learning Intention
Success Criteria

1. Use the appropriate form of Pythagoras Theorem to
   solving harder problems.

1. To use knowledge already gained on Pythagoras
Theorem to solve harder problems using Theorem
twice.
2. Show all working.
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30
Solving Real-Life Problems
S3 Credit
Problem Find the length of h.
Find length BD first
h
B
C
15
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12
A
D
13
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Solving Real-Life Problems
S3 Credit
Problem Find the length of length y.
Now find h
h
B
C
15
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12
A
D
13
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Solving Real-Life Problems
S3 Credit
Problem Find the diagonal length of the cuboid
AG.
Find AH first
F
G
B
C
7cm
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E
H
6cm
10cm
A
D
8cm
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Solving Real-Life Problems
S3 Credit
Problem Find the diagonal length of the cuboid
AG.
Now find AG
F
G
B
C
7cm
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E
H
6cm
10cm
A
D
8cm
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Pythagoras Theorem
S3 Credit
Now try Ex 5.1 Ch8 (page 154)
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S3 Starter Questions
S3 Credit
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Converse of Pythagoras Theorem
S3 Credit
Learning Intention
Success Criteria

• Apply the converse of Pythagoras Theorem to prove
  a triangle is
• right-angled.

1. To explain the converse of Pythagoras Theorem
to prove a triangle is right-angled.
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Converse1 talk Converse2 opposite, reverse
Converse of Pythagoras Theorem
S3 Credit
c
Converse Theorem states that if
b
a
1. Then triangle MUST be right-angled.
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2. Right-angle is directly opposite C.
Hypotenuse
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Converse of Pythagoras Theorem
S3 Credit
Problem Is this triangle right-angled ? Explain
Answer
If it is then Pythagoras Theorem will be true
10cm
6 cm
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9 cm
By the Converse Theorem, triangle is NOT
right-angled
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Converse of Pythagoras Theorem
S3 Credit
Problem A picture frame manufacturer claims
that his are rectangular is his claim true.
If it is then Pythagoras Theorem will be true
50cm
40 cm
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30 cm
By the Converse Theorem, frame IS rectangular
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Converse of Pythagoras Theorem
S3 Credit
Now try Ex 6.1 6.2 Ch8 (page 156)
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