Math Tutorial 1

The Pythagorean Theorem was created by Pythagorus, a greek philosopher that lived from 580 to 500 BC. His theorom states that the sum of the two sides of a right triangle, both squared , will equal the square of the hypotenuse. The section below will demonstrate through geometric proof one of the ways in which this formula can be proven.

figure 1.2

In figure 1.2 the two sides of the right angle re labeled a and b with the hypotenuse labeled c. This right triangle will be used to prove that a² + b² = c². The shape shown in figure 1.3 consists of the original triangle and three right triangles rotated to 90^o, 180^o, and 270^o degrees from the origin of the original triangle. Each triangle has an area of Length * Width divided by two or ( a*b/2 )

figure 1.3

From geometry we know that the area of a square is the length of one of its sides squared. In figure 1.3 the length of the sides of the inner square can be determined by subtracting the length of A from the length of b, therefore the area of the outer square is equal to c²and the area of the inner square is equal to (a-b)².

Now we pull all this information together; The area of the outer square is equal to the area of the inner square plus the area all 4 triangles.

c² = (a-b)² + (4*((a*b)/2))

c² = (a-b)² + 2ab

c² = a² - 2ab + b² + 2ab

c² = a² + b²

This look at the pythagorean theorem should not only explain in more detail how it actually works, but should give more insight into the possible applications of the theorem. It has been proven that, for a right triangle, the hypotenuse squared is equal to the sum of the squares of the sides. This can be extremely useful in any situation where two of the sides are given and it is necessary to find the length of the third side. For example:

In addition, the variation of the formula that solves for the hypotenuse is the foundation for the distance formula.

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