The Pythagorean Theorem Essay Example for Free

The Pythagorean Theorem EssayIn the world of mathematics, the Pythagorean Theorem is whizz of the most popular theorems and is widely applied in many problems and applications because of its basic and simple concept. It is a likeness in Euclidean geometry relating the three stances of a even out triangle. The theorem is named after the Greek mathematician and philosopher, Pythagoras, who lived in the 6th speed of light B.C. It is one of the earliest theorems known since the ancient civilizations.The Pythagorean Theorem states thatIn any securefield angle triangle, the theater of the squ are of the status opposite the right angle i. e. whose side is the hypotenuse is equal to the correspond of the areas of the even ups of the two sides that project at a right angle i.e. whose sides are the two legs.In another(prenominal) words,The strong on the hypotenuse is equal to the sum of the squares on the other two sides.Consider a right triangle first rudiment with right a ngle at A.BAC = 90 degreesThen, the square drawn on BC opposite the right angle is equal to the two squares unneurotic on BA and AC. Thus, the sides of a right triangle are related by the squares drawn on them.The Pythagorean Theorem is a statement about triangles containing a right angle. It states thatThe area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.Illustratation by numbersLet the sides of the right angle triangle be 3, 4, and 5. Then the square drawn on the side opposite the right angle is 25, which is equal to the squares on the sides that make the right angle 9 + 16. The side opposite the right angle is called the hypotenuse. Thus the theorem can be expressed as the equating 32 + 42 = 52.This proves the earlier statement which isThe square on the hypotenuse is equal to the sum of the squares on the other two sides.ProofsThis is a theorem that may have more known proofs than any other .Consider a right triangle with sides a, b, and c as hypotenuse.Let a, b, and c arrange four of those triangles to form a square whose side is a+b as shown above in Fig.1. Now, the area of that square is equal to the sum of the four triangles, plus the interior square whose side is c.Two of those triangles taken together, however, are equal to a rectangle whose sides are a, b. The area of such a rectangle is a times b ab. Therefore the four triangles together are equal to two such rectangles. Their area is 2ab.As for the square whose side is c, its area is manifestly c. Therefore, the area of the entire square isc + 2ab . . . . . .(1)At the same time, an equal square with side a + b (Fig. 2) is made up of a square whose side is a, a square whose side is b, and two rectangles whose sides are a, b. Therefore the area of that square isa + b + 2abBut this is equal to the square formed by the triangles, line(1)a + b + 2ab = c + 2ab.Therefore, on subtracting the two rectangles 2ab from each square, we are left witha + b = c.This is the Pythagorean TheoremWorks CitedBell, John L. The Art of the Intelligible An principal(a) Survey of maths in its Conceptual Development. USA Kluwer, 1999.Dunham, W. Euclids Proof of the Pythagorean Theorem. Journey through whizz The Great Theorems of Mathematics. New York Wiley, 1990.Maor, Eli. The Pythagorean Theorem A 4,000-Year History. Princeton. New Jersey Princeton University Press, 2007.Morris, Stephanie J. The Pythagorean Theorem. 2008. The University of Georgia Department of Mathematics Education. 1 May 2008 http//jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html.Spector, Lawrence. The Pythagorean Thoerem. The Math Page. 2008. 30 April 2008 http// www.themathpage.com/aTrig/pythagorean-theorem.htm.Weisstein, Eric W. Pythagorean Theorem. MathWorld. 1 May 2008. Wolfram blade Resource. 3 May 2008 http//mathworld.wolfram.com/Pythagorean Theorem.html.