Pythagorean Theorem

Subject: Pythagorean Theorem Sat Jun 28, 2008 7:13 pm

Pythagorean theorem
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The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).In mathematics, the Pythagorean theorem (American English) or Pythagoras' theorem (British English) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof,[1] although knowledge of the theorem almost certainly predates him. The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

This is usually summarized as follows:

The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation:

or, solved for c:

If c is already given, and the length of one of the legs must be found, the following equations can be used (The following equations are simply the converse of the original equation):

or

This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle it reduces to the Pythagorean theorem.
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Subject: Re: Pythagorean Theorem Sat Jun 28, 2008 7:14 pm

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Pythagorean theorem
From Wikipedia, the free encyclopedia
(Redirected from Pythagorean theorum)
Jump to: navigation, search

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).In mathematics, the Pythagorean theorem (American English) or Pythagoras' theorem (British English) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof,[1] although knowledge of the theorem almost certainly predates him. The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

This is usually summarized as follows:

The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation:

or, solved for c:

If c is already given, and the length of one of the legs must be found, the following equations can be used (The following equations are simply the converse of the original equation):

or

This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle it reduces to the Pythagorean theorem.

Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.Trigonometry
History
Usage
Functions
Inverse functions
Further reading

Reference
List of identities
Exact constants
Generating trigonometric tables
CORDIC

Euclidean theory
Law of sines
Law of cosines
Law of tangents
Pythagorean theorem

Calculus
The Trigonometric integral
Trigonometric substitution
Integrals of functions
Integrals of inverses

Contents [hide]
1 History
2 Proofs
2.1 Proof using similar triangles
2.2 Euclid's proof
2.3 Garfield's proof
2.4 Similarity proof
2.5 Proof by rearrangement
2.6 Algebraic proof
2.7 Proof by differential equations
2.8 Rational trigonometry
3 Converse
4 Consequences and uses of the theorem
4.1 Pythagorean triples
4.2 List of primitive Pythagorean triples up to 100
4.3 The existence of irrational numbers
4.4 Distance in Cartesian coordinates
5 Generalizations
5.1 The Pythagorean theorem in non-Euclidean geometry
6 Cultural references to the Pythagorean theorem
7 See also
8 Notes
9 References
10 External links

[edit] History
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The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, knowledge of the relationship between adjacent angles, and proofs of the theorem.

Megalithic monuments from circa 2500 BC in Egypt, and in Northern Europe, incorporate right triangles with integer sides.[2] Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically.[3]

Written between 2000 and 1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem whose solution is a Pythagorean triple.

During the reign of Hammurabi the Great, the Mesopotamian tablet Plimpton 322, written between 1790 and 1750 BC, contains many entries closely related to Pythagorean triples.

The Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th century BC and the 2nd century BC, in India, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle.

The Apastamba Sulba Sutra (circa 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert Bŭrk, this is the original proof of the theorem; he further theorizes that Pythagoras visited Arakonam, India, and copied it.

Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proklos's commentary on Euclid. Proklos, however, wrote between 410 and 485 AD. According to Sir Thomas L. Heath, there is no attribution of the theorem to Pythagoras for five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.[4]

Around 400 BC, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Circa 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.

Written sometime between 500 BC and 200 AD, the Chinese text Chou Pei Suan Ching (周髀算经), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives a visual proof of the Pythagorean theorem — in China it is called the "Gougu Theorem" (勾股定理) — for the (3, 4, 5) triangle. During the Han Dynasty, from 202 BC to 220 AD, Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles.[5]

The first recorded use is in China, known as the "Gougu theorem" (勾股定理) and in India known as the Bhaskara Theorem.

There is much debate on whether the Pythagorean theorem was discovered once or many times. Boyer (1991) thinks the elements found in the Shulba Sutras may be of Mesopotamian derivation.[6]
_________________
In Memory of the Bloo Poo, the Turd of Tethys.
Status: K.I.A.
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We Will Miss You, and you Bloo-ness.

Subject: Re: Pythagorean Theorem Sat Jun 28, 2008 7:15 pm

Too much crap about the theorum. I WILL NOT POST IT.
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In Memory of the Bloo Poo, the Turd of Tethys.
Status: K.I.A.
Lifespan: March 23, 2008- June 6,2008
We Will Miss You, and you Bloo-ness.

CHICKEN Joined : 21 Feb 2008 Posts : 689

fish and fat Russian midgets
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