When I first heard the Pythagorean Formula, I wondered where did Pythagoras get the idea of that theorem? My teacher showed a right triangle and drew squares upon each side. That was so unintuitive to me. It was a deduction after the fact. It didn’t motivate my understanding of the formula.
Years later, I stumbled across a construction that made much more sense. Consider a square. Divide a side anywhere into two parts, name them, a and b. Then from ‘a’ draw a straight line to ‘a’ on the adjacent side. Do the same to the third side, the fourth side, then close the inscribed square by drawing a straight line from the fourth side to the original point, a.
Figure 1 shows the original square of side a+b and the inscribed square of side c.
The area of the outer square can be calculated two ways, as the outer square or as the inner square plus the four triangles.
Area of original square = Area of inscribed square + 4* identical right triangles
Area of outside square, (side a + b) = area of inside square, (side c) + 4 times area of the right triangle, (altitude a, base b)
(a + b)2 = c2 + 4(½ ab)
a2 + 2ab + b2 = c2 + 2ab
a2 + b2 = c2