Use the Pythagorean theorem to find the length a in the figure. Then find b, c, d, e, f,...

Use the Pythagorean theorem to find the length a in the figure. Then find b, c, d, e, f, and g.

Note: This figure provides a geometric construction for the irrational numbers , where n is a nonsquare natural number. According to Boyer’s A History of Mathematics, 2nd ed. (New York: John Wiley & Sons, Inc., 1991), “Plato . . . says that his teacher Theodorus of Cyrene . . . was the first to prove the irrationality of the square roots of the nonsquare integers from 3 to 17, inclusive. It is not known how he did this, nor why he stopped with One plausible reason for Theodorus’s stopping with may have to do with the figure shown here. Theodorus may have known that the figure begins to overlap itself at the stage where would be constructed.