A Proof of the Pythagorean Theorem

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A Proof of the Pythagorean Theorem
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• A Proof of the
• Pythagorean
• Theorem

Do not use page arrows. Just patiently click
your way through the presentation.
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First we will set things up. Begin by drawing a
square.
a
b
Next, place a point anywhere on the top side.
Label the length from one corner to the point a.
Label the length from the other corner to the
point b.
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a
b
Using the same lengths a and b, place a point on
each side as shown.
a
b
In terms of a and b, what is the length of a side
of the square?
b
a
a b
a
b
What is the area of the square?
(a b) (a b) or (a b)2
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Now connect the points to form four congruent
right triangles.
c
c
Label each hypotenuse c.
c
c
The hypotenuses of the triangles form a square.
How do you know that this is a square?
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What is the area of the inside square?
c2
What is the area of each triangle?

ab
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Now we are set up to provide a proof of the
Pythagorean Theorem.
Note that the area of the inner square
is equal to
the area of the outer square
minus the area of the four triangles.
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Remember that the area of the inner square is
c2.
The area of the outer square is
(a b)2.
The area of the four triangles combined is
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ab) 2ab.
4(
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Area inner square Area outer square - Area of 4
triangles
c2 (a b)2 - 2ab
c2 (a b) (a b) - 2ab
c2 a2 2ab b2 - 2ab
c2 a2 b2
The Pythagorean Theorem!
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The Pythagorean Theorem
c2 a2 b2
This proof shows that the Pythagorean Theorem
holds for any right triangle. How can we be sure
of this?
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