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PYTHAGORA'S THEOREM

INTRODUCTION

I drew a right triangle on the board with squares on the hypotenuse and legs and observed the fact the the square on the hypotenuse had a larger area than either of the other two squares. Then I asked, "Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. Which would you choose?" Interestingly enough, about half the class opted for the one large square and half for the two small squares. Both groups were equally amazed when told that it would make no difference.

THE THEOREM

The Pythagorean (or Pythagoras') Theorem is the statement that the sum of (the areas of) the two small squares equals (the area of) the big one.

In algebraic terms, a² + b² = c² where c is the hypotenuse while a and b are the legs of the triangle. 

 

     File:Pythag anim.gif                                       

OBSERVATIONS

The theorem is of fundamental importance in Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got thoroughly forgotten.

REMARK

The statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. The Theorem is reversible which means that its converse is also true. The converse states that a triangle whose sides satisfy a² + b² = c² is necessarily right angled. Euclid was the first (I.48) to mention and prove this fact

APPLICATIONS

Pythagorean triples

A Pythagorean triple has three positive integers a, b, and c, such that a2 + b2 = c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.[1] Evidence from megalithic monuments in Northern Europe shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).

A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1).

The following is a list of primitive Pythagorean triples with values less than 100:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Algebraic proofs

 
Diagram of the two algebraic proofs

The theorem can be proved algebraically using four copies of a right triangle with sides a, b and c, arranged inside a square with side c as in the top half of the diagram.[16] The triangles are similar with area \tfrac12ab, while the small square has side ba and area (ba)2. The area of the large square is therefore

(b-a)^2+4\frac{ab}{2} = (b-a)^2+2ab = a^2+b^2. \,

But this is a square with side c and area c2, so

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

A similar proof uses four copies of the same triangle arranged symmetrically around a square with side c, as shown in the lower part of the diagram.[17] This results in a larger square, with side a + b and area (a + b)2. The four triangles and the square side c must have the same area as the larger square,

(b+a)^2 = c^2 + 4\frac{ab}{2} = c^2+2ab,\,

giving

c^2 = (b+a)^2 - 2ab = a^2 + b^2.\,
 

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