The Pythagorean Theorem

In the world of mathematics, the Pythagorean Theorem is one of the most popular theorems and is widely applied in many problems and applications because of its basic and simple concept. It is a relation in Euclidean geometry relating the three sides of a right triangle. The theorem is named after the Greek mathematician and philosopher, Pythagoras, who lived in the 6th century B. C.  It is one of the earliest theorems known since the ancient civilizations.

The Pythagorean Theorem states that:

“In any right angle triangle, the area of the square  of  the side opposite the right angle i.

e. whose side is the hypotenuse  is equal to the sum of the areas of the squares of the two sides that meet at a right angle i.e. whose sides are the two legs.”

In other words,

“The square on the hypotenuse is equal to the sum of the squares on the other two sides.”

Consider a right triangle ∆ABC with right angle at A.

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BAC = 90 degrees

Then, the square drawn on BC opposite the right angle is equal to the two squares together 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 that:

"The 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 numbers

Let 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 equation: 3² + 4²   = 5².

This proves the earlier statement which is

“The square on the hypotenuse is equal to the sum of the squares on the other two sides.”

Proofs

This 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 simply c².  Therefore, the area of the entire square is

c² + 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 is

a² + b² + 2ab

But this is equal to the square formed by the triangles, line (1):

a² + b² + 2ab = c² + 2ab.

Therefore, on subtracting the two rectangles -- 2ab -- from each square, we are left with

a² + b² = c².

This is the Pythagorean Theorem

Works Cited

Bell, John L. The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development. USA: Kluwer, 1999.

Dunham, W. "Euclid's Proof of the Pythagorean Theorem." Journey through Genius: 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 Web Resource. 3 May 2008 <http://mathworld.wolfram.com/Pythagorean Theorem.html>.