Pythagorean Theorem

Theory 1

Pythagorean Theorem

The Pythagorean Theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof. The concept of this mathematical theory is, in any right triangle, the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares whose sides are the two legs the two sides that meet at a right angle. In the math world there are dozens of proofs that make this specific theory a fact. In this paper I will show you five and how the equations work.

The first is the proof by, "Professor R. Smullyan in his book 5000 B.C. and Other Philosophical Fantasies". In this theory he uses the equation a² + b² = c² where c is the hypotenuse while a and b are the legs of the triangle. A example is here in this drawing of the problem.

Another proof is using the equation, "a² = cx and b² = c(c - x)". The example of the equation in picture form is here.

The third proof is by "Euclid VI.31 in translation by Sir Thomas L. Heath". It states that Let ABC be a right-angled triangle having the angle BAC right; I say that the figure on BC is equal to the similar and similarly described figures on BA, AC. Let AD be drawn perpendicular. Then since, in the right-angled triangle ABC, AD has been drawn from the right angle at A perpendicular to the base BC, the triangles ABD, ADC adjoining the perpendicular are similar both to the whole ABC and to one another [VI.8]. And, since ABC is similar to ABD, therefore, as CB is to BA so is AB to BD.

The fourth proof of the theorem says that by drawing a circle with radius c and a right triangle with sides a and b as shown. In this situation, one may apply any of a few well known facts. For example, in the diagram three points F, G, H located on the circle form another right triangle with the altitude FK of length a. Its hypotenuse GH is split in two pieces: (c + b) and (c - b). So in conclusion you get a² = (c + b)(c - b) = c² - b².

The fifth and last theorem states that in the example picture below there are several similar triangles (abc, a'b'c', a'x, and b'y.) We successively have y/b = b'/c, x/a = a'/c, cy + cx = aa' + bb'. And, finally, cc' = aa' + bb'.

In conclusion about this well used math theorem I have learned a lot while doing the research for this paper. There were so many choices to choose