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Revision as of 04:08, 14 December 2002 by AxelBoldt (talk | contribs)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)The Pythagorean theorem or Pythagoras' theorem is named for and attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, though the facts of the theorem were known before he lived. It states a relationship between the areas of the squares on the sides of a right triangle:
- The sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse.
(A right triangle is one with a right angle; the legs are the two sides that make up the right angle; the hypotenuse is the third side opposite the right angle; the square on a side of the triangle is a square one of whose sides is that side of the triangle).
| Image of a right triangle to illustrate the Pythagorean Theorem |
Since the area of a square is the square of the length of a side, we can also formulate the theorem as:
- Given a right triangle, with legs of lengths a and b and hypotenuse of length c, then
- a+b=c.
Proof: Draw a right triangle with sides a, b, and c as above. Then take a copy of this triangle and place its a side in line with the b side of the first, so that their c sides form a right angle (this is possible because the angles in any triangle add up to two right angles -- think it through). Then place the a side of a third triangle in line with the b side of the second, again in such a manner that the c sides form a right angle. Finally, complete a square of side (a+b) by placing the a side of a fourth triangle in line with the b side of the third. On the one hand, the area of this square is (a+b) because (a+b) is the length of its sides. On the other hand, the square is made up of four equal triangles each having area ab/2 plus one square in the middle of side length c. So the total area of the square can also be written as 4 · ab/2 + c. We may set those two expressions equal to each other and simplify:
- (a+b) = 4 · ab/2 + c
- a + 2ab + b = 2ab + c
- a + b = c
The sheer volume of distinct known proofs of this theorem is staggering. See http://www.cut-the-knot.com/pythagoras/index.html for just a sampling.
The converse of the Pythagorean Theorem is also true: if you have three positive numbers a, b, and c such that a+b=c, then there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides a and b. This can be proven using the law of cosines which is a generalization of the Pythagorean theorem applying to all triangles, not just right-angled ones.
Another interpretation of the Pythagorean Theorem was already given by Euclid in his Elements: if one erects similar figures (see Geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.
Yet another generalization of the Pythagorean Theorem is Parseval's identity in inner product spaces.
The Pythagorean Theorem also generalizes to higher-dimensional simplexes. If a tetrahedron has a right angle corner (a corner of a cube), then the square of the area of the hypotenuse is the sum of the squares of the other three faces. This is called de Gua's theorem.
One should note that the Pythagorean Theorem is a theorem that follows from the axioms of Euclidean geometry and does not apply to triangles in non-Euclidean spaces such as the surface of a sphere. A priori, it therefore need not be true for actually existing triangles that we measure out in our Universe. One of the first mathematicians to realize this was Carl Friedrich Gauss, who actually carefully measured out large right triangles as part of his geographical surveys in order to check the theorem. He found the theorem to hold. The theory of general relativity holds that matter and energy causes space to be non-Euclidean and the theorem does not apply, although the deviation from Euclidean space is small except near strong gravitational sources such as black holes. Whether the theorem is violated over large cosmological scales is an open problem of cosmology.
See also: