The Pythagorean Theorem

        The Pythagorean Theorem claims that a²+b²=c² where c is the hypotenuse while a and b are the other two sides of the right triangle.  the area of the two small squares equals the area of the large one. 

         Pythagoras, for whom the famous theorem is named, lived during the 6th century B.C. on the island of Samos in the Aegean Sea. He also lived in Babylon and in southern Italy. Pythagoras was a teacher, a philosopher, a mystic and, to his followers, almost a god.  Pythagoras of Samos is often described as the first pure mathematician.   He is an extremely important figure in the development of mathematics yet we still know relatively little about his mathematical achievements. Unlike many later Greek mathematicians that left some of the books that they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which, certainly means that today Pythagoras is a mysterious figure. 

        Little is known of Pythagoras's childhood. All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark which Pythagoras had on his thigh. It is probable that he had two brothers although some sources say that he had three. Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer . There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras.

        The Pythagorean Theorem is very important in mathematics, but also in real-life.  In math, it helps us find the length of an unknown side of a triangle, which can be very useful.  Outside of the classroom it can be used to find the distance from one point to     another. 

Example:    *Joe wants to find out the actual distance from 1st base to 3rd base.  He knows that it is 90ft from 1st to 2nd  and another 90ft from 2nd to 3rd.

            90² = 8100                            8100 + 8100 = 16200²ft

            90²= 8100                                                                                    ²root of 16200 = 127.28ft

    *After using the Pythagorean Theorem to solve his problem John calculated that the length in feet between 1st and 3rd base was about 127ft.

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A Pythagorean Theorem Proof

We start with two squares with sides a and b, respectively, placed side by side. The total area of the two squares is a²+b².  The construction did not start with a triangle but now we draw two of them, both with sides a and b and hypotenuse c. Note that the segment common to the two squares has been removed. At this point we therefore have two triangles and a strange looking shape.  As a last step, we rotate the triangles 90^o, each around its top vertex. The right one is rotated clockwise whereas the left triangle is rotated counterclockwise. Obviously the resulting shape is a square with the side c and area c².

References

 Bogomolny, Alexander.  Cut-the-Knot.  http://www.cut-the-knot.com/ctk/index.shtml : Internet.  1996-2000

Morris, Stephanie J.  The Pythagorean Theorem.  http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html : Internet.  Unknown Date

Jansson, Jane.  Interview.  2003

-                                                                                                                                            by George Jansson (2003)