The Pythagorean Theorem

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The Pythagorean Theorem
a
c

b

2
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Created by Natalie Arritt, 2006
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Table of Contents

• Who is Pythagoras?
• What is the Pythagorean Theorem?
• When do you use the Pythagorean Theorem?
• How do you prove the Pythagorean Theorem?
• Four different proofs of the Pythagorean Theorem.

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Who is Pythagoras?
http//www.newgenevacenter.org/reference/contents.
htm

• Pythagoras was a mathematician who lived from
  560-480 BC. He is well know for his school. He
  allowed all to come to his school including
  females, which was unusual for his time. His
  students were placed under a strict code of
  conduct and thought.

http//en.wikipedia.org/wiki/Pythagoras
There are arguments about who actually first
proved the Pythagorean Theorem. There is no
evidence of whether Pythagoras or one of his
students actually developed the theorem.
http//home.c2i.net/greaker/comenius/9899/pythagor
as/pythagoras.html
What is the Pythagorean Theorem?
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What is the Pythagorean Theorem?

• The Pythagorean Theorem is a theorem about the
  sides of a right triangle.
• It says that in any right triangle, the sum of
  the square of the legs is equal to the square of
  the hypotenuse.

• We call the legs a and b.
• The hypotenuse is called c.

Hypotenuse
c
a
Leg

• The Theorem is written in equation form.
• a2 b2 c2

Leg
b

• Pythagorean song (by Harry Guffee)

http//www.songsforteaching.com/guffee/pythagorean
s.htm
When do you use the Pythagorean Theorem?
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When do you use the Pythagorean Theorem?

• The Pythagorean Theorem is most comely used to
  find the sides of right triangles.
• If you are given two sides of any right triangle
  you can find the third side.

• In this triangle we can find side with length, x,
  by using the Pythagorean Theorem.
• First substitute what you know into the equation
  a2 b2 c2.

• a 3, b 4 and c x. So we have 32 42
  x2.
• Now simplify, 9 16 x2
• 25 x2
• 
  5 x c

v
v
How do you prove the Pythagorean Theorem?
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How do you prove the Pythagorean Theorem?

• To prove the Pythagorean Theorem you need to
  show that a2 b2 c2.

c
a
b
See four different proofs
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Four proofs of the Pythagorean Theorem

• Proof 1(Euclids First Proof)
• Proof 2
• Proof 3 (Garfields Proof)
• Proof 4 (Similar Triangle Proof

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the Pythagorean Theorem?
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Euclids first proof
c

• First start with a right triangle.
• Second make squares the same length as each side
  of the triangle.
• Now move a2, so that it fills in part of c2.
• Next cut b2 into smaller squares.
• The final step is to fill in the rest of c2 with
  the new squares from b2.
• Now you can see that a2 b2 c2.

c
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the Pythagorean Theorem
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See Proof 2
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Proof 2

• Start with four congruent right triangles.
• Each triangle has an area of ½(ab).
• Now we move these triangles to form a square
  with sides c.
• This big square has a smaller square in it with
  sides a-b.
• The area of the big square is c2.
• The area of the big square is also equal to the
  sum of the four triangles and the small square.
  (a-b)4(½(ab)) .
• By using substitution we get
  c2 (a-b)2 4(½(ab)) .

• Now simplify the equation.
• c2 (a-b)2 2ab
• c2 a2 - 2ab b2 2ab
• So c2 a2 b2

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Back to Proof 1
See Proof 3
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Garfields Proof

• So by substitution,
  ½(ab)(ab) 2( ½ab)(½ c2 )
• Now simplify
  
• a2 2ab b2 2ab c2
• Now you can see that a2 b2 c2

• Start with a two congruent right triangles. Each
  with area ½(ab).
• Connect them together so that you have a total
  side ab.
• Draw in line segment XY opposite side ab.
• Now we have formed another triangle with sides
  c, c, and XY. With area ½(c2 ).
• Now we have a trapezoid. The area of the
  trapezoid is equal to ½ the sum of the bases
  times the height.( ½(ab)(ab) )
• The area is also equal to the sum of the three
  triangles. ( 2( ½ab)(½ c2 ) )

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Back to Proof 2
See Proof 4
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Similar Triangle Proof

• Draw a right triangle with sides a, b, c and
  angles A, B, C.
• Now Draw in the altitude, h, from C.
• This creates two new triangles with a, h, x and
  h, b, y. Where xy c.
• So now we have three similar triangles by AA.
• So by corresponding sides of similar triangles
  we get a c and b c

• Cross multiply to get a2 cx and b2 cy.
• By equivalent rules of addition
  a2 b2 cx cy
• c(xy)
• c(c) (by substitution)
• c2
• Now you can see that a2 b2 c2

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Back to Proof 3
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Teacher Page

• Lesson Plan
• Worksheet
• References
• Technology Portfolio Table of Contents

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References
http//www.songsforteaching.com/guffee/pythagorean
s.htm
http//www.newgenevacenter.org/reference/contents.
htm
http//en.wikipedia.org/wiki/Pythagoras
http//home.c2i.net/greaker/comenius/9899/pythagor
as/pythagoras.html