The Pythagorean theorem states
that
the sum of the surface areas generated by the sides of
a right
triangle,
is equal to the surface area generated by the the
hypotenuse. Traditionally the example used to illustrate the Pythagorean theorem is the 3-4-5 diagram. This is a fallacy, and the purpose of this message it show why it is a fallacy. The formula for the Pythagorean theorem can be stated as follows: (a sq + 1) + ( b sq +1) = (c sq + 1) The incorrect formula omits the +1 for each value. It simply states that: a sq + b sq = c sq. The addition of one for each value is necessary in order to convert the area, to a surface area. In reality there can be no area without a surface. The "one" that is added, makes the area real, it adds the third dimension of substance, to the formula. From a practical point of view it can be seen as one square area being overlapped when we multiply length by width. The 3-4-5 formula is an approximation and was apparently used by ancient builders in order to make sure that everything was 'square'. The sketch below shows how the 3-4-5 formula distorts the right angle, making it acute(less than 90 degrees). Notice that the only linear deviation, is the length EB, which is extended beyond 5 units in order to accommodate the distortion caused by the faulty formula. Needless to say all the angles are distorted. In reality the lengths AB and BC, should each be 5 1/10 to establish a right angle at D. No geometric rules have been violated and because the angle y, is small, the deviation is subtle. |