1. various visualizations
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 b a a c b c a+b c b c a a+b a b Area + + + +
must be equal
Thus,

4. Pythagorean Theorem, IIb a 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, IIIb
b-a c b
b-a 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