Pythagorean Theorem
various visualizations
Pythagorean Theorem
• If this was part of a face-to-face lesson, I would cut out four right
triangles for each pair of participants and ask you to discover these
visualizations of why the Pythagorean Theorem is true.
• Before you begin you might want to cut out four right triangles and
play along!
Pythagorean Theorem, I
b
a
a
c
b
c
a+b
c
b
c
a+b
a
Area  ( a  b ) * ( a  b )
2
 a  ab  ba  b
2

2
Thus,
2
a b c
2
2
1
a*b 
 2 ab  c
a  2 ab  b  2 ab  c
2
+
2
must be
equal
2
b
+
Area =
 a (a  b)  b(a  b)
 a  2 ab  b
a
2
2
1
2
2
a*b 
1
2
+
a*b 
+
1
2
a *b  c
2
Pythagorean Theorem, II
a
b
a
a
a
c
b
c
b
b
b
c
c
a
a
b
Notice that each square has 4 dark green triangles.
Therefore, the yellow regions must be equal.
Yellow area
 a b
2
2
Yellow area
 c
2
Pythagorean Theorem, III
aa
b
b-a
b
c
b-a
c
a
c
c
Area of whole square
Area of whole square
 c*c  c
2
 a rea o f 4 g reen tria n g les  a rea o f w h ite sq u a re
1
 4
2
must be
equal

a b   ( b  a )( b  a )

 2 ab  b (b  a )  a (b  a )
2
 2 ab  b  ab  ab  a
2
 2 ab  b  2 ab  a
2
 b a
2
2
2
Pythagorean Theorem
• The next demonstration of the Pythagorean Theorem involve cutting
up the squares on the legs of a right triangle and rearranging them
to fit into the square on the hypotenuse. This demonstration is
considered a dissection.
• I highly recommend paper and scissors for this proof of the
Pythagorean Theorem.
Pythagorean Theorem, IV
•
•
Construct a right triangle.
Construct squares on the sides.
•
Construct the center of the square on the
longer leg. The center can be constructed
by finding the intersection of the two
diagonals.
•
Construct a line through the center of the
square and parallel to the hypotenuse.
Pythagorean Theorem, IV
•
Construct a line through the center of the
square and perpendicular to the
hypotenuse.
•
Now, you should have four regions in the
square on the longer leg. The five interiors:
four in the large square plus the one small
square can be rearranged to fit in the square
on the hypotenuse. This is where you will
need your scissors to do this.
•
Once you have the five regions fitting inside
the square on the hypotenuse, this should
2
2
2
illustrate that a  b  c
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Pythagorean Theorem