Pythagorean Theorem

1
Pythagorean Theorem

• various visualizations

2
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!

3
Pythagorean Theorem, I
a
b
a
c
b
c
ab
c
b
c
a
ab
a
b
Area




Area
must be equal
Thus,
4
Pythagorean Theorem, II
a
a
b
a
a
c
b
c
b
c
b
b
c
a
a
b
Notice that each square has 4 dark green
triangles. Therefore, the yellow regions must be
equal.
Yellow area
Yellow area
5
Pythagorean Theorem, III
b
a
a
b-a
b-a
b
c
c
a
c
c
Area of whole square
Area of whole square
must be equal
6
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.

7
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.

8
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 illustrate
  that