Mike's mental image of a hinge closing to a zero angle and opening to a straight angle was helpful to me. Start side a at the origin extending along the x axis to the right. Start side b where a ends. Theta = 0, c = a+b, c^2 = a^2 + 2ab + b^2. Now rotate b about the "hinge" where a joins b and set theta = pi, c = b - a, c^2 = (b - a)^2 = a^2 - 2ab + b^2. Calculate the midpoint: theta = pi/2, c^2 = a^2 + b^2. Alternatively, consider that for theta = 0, (a + b)^2 > a^2 + b^2. As theta increases as b rotates about the hinge, c = a + b cos theta < a + b cos 0. We want c small enough to get rid of 2ab in (a + b)^2, but no smaller. a^2 + b^2 = (a + b cos theta)^2. Thus c = a + b cos theta = Sqrt( a^2 + b^2). Jeff On Mon, Feb 15, 2016 at 3:00 PM, Mike Beeler <mikebeeler@verizon.net> wrote:
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
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