If we want a motivation for why one would seek something like the Pythagorean theorem, how about… Take a right triangle with one leg horizontal and fixed in place. Replace the right angle with a hinge, and the hypotenuse with a rubber band. Opening the hinge to a straight angle and drawing squares on each side, the hypotenuse’s square is seen to be bigger than the sum of the other two. Closing the hinge to a zero angle and again drawing squares, the hypotenuse’s square is seen to be smaller than the sum of the other two. So there must be some angle where they are equal, but what is the angle? Investigation discovers it is a right angle, for any length of the legs. Why draw squares and not cubes or just the side lengths themselves? Probably you do try those, but nothing interesting comes of it. Math as engineering, tinkering around, noticing when something useful happens. — Mike