Oreški Ivana 779 Osijek 2010..  an Ionian Greek philosopher and founder of the religious movement called Pythagoreanism  developed mathematics, astronomy,

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Presentation transcript:

Oreški Ivana 779 Osijek 2010.

 an Ionian Greek philosopher and founder of the religious movement called Pythagoreanism  developed mathematics, astronomy, and music  best known for the Pythagorean theorem, which bears his name

 first introduced by Babylonian 1000 years earlier before Pythagoras did that  proved by Pythagoras  created chaos in ancient Greece

 a relation in Euclidean geometry among the three sides of a right triangle  Pythagoras explains the theorem by demonstrating a right triangle

 If and only if we have a right-angled triangle then the square of the hypotenuse is equal to the sum of the squared sides  In terms of areas:

PYTHAGOREAN TRIPLES  A Pythagorean triple has three positive integers: a, b, c.  Such a triple is commonly written (a, b, c)  The following is a list of primitive Pythagorean triples with values less than 50: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41

PYTHAGOREAN TRIGONOMETRIC IDENTITY  The sine: The cosine:

 From that it follows:  This relation between sine and cosine sometimes is called the fundamental Pythagorean trigonometric identity.

 The main use of Pythagorean theorem is trigonometry: calculating :  height of triangles,  diagonals of square,  space diagonals of cube  angle values of each triangle  Projecting buildings, distance measurement

 John leaves school to go home. He walks 6 blocks North and then 8 blocks west. How far is John from the school?

 Answer: c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 100 c = √100 c = 10 The distance from school to home is 10 blocks.

 A 13 meter ladder is placed 5 meters away from a wall. The distance from the ground straight up to the top of the wall is 13 meters. Will the ladder reach the top of the wall?

 Answer : c 2 = a 2 + b = b = 25 + b 2 b 2 = = b 2 b = √144 = 12 The ladder will never reach the top since it will only reach 12 meters high from the ground yet the top is 13 meters high.