PythagoreanTheorem

The PythagoreanTheorem is attributed to the Greek philosopher and mathematician Pythagoras who lived in the 6th century B.C. The theorem states a relationship between the lengths of the sides of a right triangle. It says:

The sum of the squares of the length the sides of a right triangle is equal to the square of the length of the hypotenuse.

Visually, the theorem can be illustrated as follows:

Given a right triangle, with sides a and b and hypotenuse c, (Figure 1)

                                                     / l

                                                   /   l

                                      c         /      l 

                                              /        l 

                                           /           l  b

                                        /              l

                                     /________________ l

                                           a

                                      Figure 1

the hypotenuse is the side opposite the right (90 degree) angle in a right triangle.

Then, c^2 = a^2 + b^2, or c = sqrt(a^2 + b^2).

Certain sets of 3 integers are useful to remember as being Pythagorean triples, that is, they are possible lengths of the sides of a right triangle. For example:

                     a                     b                       c

                     3                     4                       5

                     9                    12                      15

The PythagoreanTheorem is an important tool in the study of TrigonometricFunctions.

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Poser: (3,4,5) is a pythagorean triplet since 3^2 + 4^2 = 5^2. Which positive integers are not part of a pythagorean triplet?

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Proof: Draw right triangle with sides a,b,and c as above. Turn an identical triangle 180 degrees and stick its c side to the original triangle's to form a rectangle of sides a and b. Draw an identical rectangle perpendicular to the first with only a corner touching. Draw squares of side a and b to connect the two rectangles into one large square of side (a+b). From this diagram, the area of the large square is (a+b)^2 = a^2 + b^2 + 4(.5ab) by the formulas for the areas of squares and triangles.

Now form a square from four of the triangles by placing the a side of one triangle in line with the b side of another, so that all four sides of the figure are (a+b). Note that within the large square is a square of side c. From this diagram, the area of the large square is (a+b)^2 = 4(.5ab) + c^2.

Since (a+b)^2=itself, a^2 + b^2 + 4(.5ab) = 4(.5ab) + c^2.

Therefore, a^2 + b^2 = c^2.

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I think a visual of the triangles involved in this proof would be very helpful- even if they are really crude like mine.