What you did so far...?
Begininning with a right triangle of sides a, b, and c, (a is the hypotenuse, b is the base, c is the height) determine its area in two ways; by the standard A = (1/2)(base)(height) and by Heron's Formula. Equate these and manipulate the resulting, dreadful equation until you have derived the theorem of Pythagoras; a^2 = b^2 + c^2.
See attached for Heron's Formula