Πυθαγόρας ὁ Σάμιος

Hyperbolic_orthogonal_dodecahedral_honeycombPythagoras of Samos, known as Pythaogras, lived in Greece circa 570 – 490 BCE

Raffaello Sanzio da Urbino depicted the philosopher and mathematician in a fresco painting, “Scuola di Atene,” or “School of Athens,” (1509), located in the Vatican.

Here, Pythagoras is a member of the surrounding pantheon, representatives of the Golden Age of Greek knowledge; Raphael’s embodiment of the Renaissance spirit and intellect.

Raphael, "School of Athens," fresco 500 cm x 770 cm, 	Apostolic Palace, Vatican City, 1509 - 1511Pythagoras is best known to have formalized a relationship among the lengths of three sides of a particular triangle in which has one angle perpendicular to another side.

Geometrically, the area of any triangle can be seen as half the area of an encompassing rectangle; here, the green shaded area is 15 square units because it is exactly half of the rectangle, which contains 30 unit squares.

triangle area

Area of any Triangle: ½( base )( height )

Through trigonometry, we can change the formula by substitution to include an angle.

Since the sine of angle C = h/b , the height, h = b Sin (C); and since the base = a,

we have a formula for the area of any triangle, given one angle and two included sides:

Area = ½ a b Sin (C)

A right triangle is a special case where only one angle = 90 degrees; the side opposite this angle is the longest side, called the hypotenuse; this easily gives us a height and a base for the triangle, so its area can be calculated without using a trigonometric ratio.

Here, a right triangle with base = 4 and height = 3, will have an area equal to half of 12 = 6.

In this right triangle, the relationship of its two legs, base = 4 and height = 3, can be used to find the third side, the hypotenuse, by squaring the legs and finding those three adjoining areas = 16, 9, and 25.

But 9 + 16 = 25 !

pythagorus

In a right triangle, if the leg sides are squared,

their sum is equal to the square of the hypotenuse:

32 + 42 = 52    therefore, the hypotenuse must be = 5

In general, this is the Pythagorean Theorem:  a2 + b2 = c2

                                                                                            

Proof of the Pythagorean Theorem:

pythagorus proof

Area of Large (Blue) Square: 

Each side is ( a + b ) therefore its area = ( a + b )2 = a2 + 2ab + b2

Also, Area of Large Square = Area of 4 Triangles plus Small (Green) Square

      a2 + 2ab + b2 = 4(1/2 ab) + c2

a2 + 2ab + b2 = 2ab + c2

  a2 + 2ab + b2 – 2ab = 2ab – 2ab + c2

  a2 + b2 = c2

                                                                                           

The following is a list of right triangles with integer solutions for all three sides, known as:

Primitive Pythagorean Triples

Integer solutions are found for all x, y, z, such that x2 + y2 = z2

Let:   x = u2 – v2                   y = 2 u v              and  z = u2 + v2

where u and v are integers, have opposite parity, and u > v > 0

The following list is complete for the first 25 possible right triangles;

they are called “primitive” since multiples of each x, y, z, are possible–

thus capable of generating further triangles.

 u                        v                                                x            y            z

   2                        1                                                 3            4            5

     3                       2                                                  5            12          13

      4                        1                                                15            8            17

       4                        3                                                 7            24            25

        5                        2                                                21           20            29

        5                        4                                                   9           40            41

        6                        1                                                35            12            37

        6                        3                                                27            36            45

        6                        5                                                11            60            61

        7                        2                                                45            28            53

        7                        4                                                33            56            65

       7                        6                                                 13            84           85

        8                        1                                                63            16            65

        8                        3                                                55            48            73

        8                        5                                                39            80            89

          8                       7                                                15           112           113

        9                        2                                                77            36            85

        9                        4                                                65            72            97

          9                        6                                                45           108           117

          9                        8                                                17           144           145

          10                        1                                               99              20           101

          10                        3                                                91            60            109

          10                        5                                                75           100           125

           10                        7                                                51           140           149

           10                        9                                                19           180           181

jensen-honor pythag

                                                                                                                                         

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