Why does a² + b² = c² work for every right triangle? Build the squares, then slide four triangles around a big square and watch the proof happen.
You solved all 8 challenges! The four triangles fill the same big square both ways, so the space they leave, c² or a² + b², must be the same. That is the Pythagorean theorem.
Build a right triangle
Set the two legs, a and b. The long side, c, is the hypotenuse.