In mathematics, the Pythagorean Theorem — or Pythagoras’ theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states: In any right-angled 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). The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
Where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides. The Pythagorean Theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof. The converse of the theorem is also true: For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b. An alternative statement is:
For any triangle with sides a, b, c, if a2 + b2 = c2, then the angle between a and b measures 90°. This converse also appears in Euclid’s Elements (Book I, Proposition 48): “If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right. ” It can be proven using the law of cosines or as follows: Let ABC be a triangle with side lengths a, b, and c, with a2 + b2 = c2.
Construct a second triangle with sides of length a and b containing a right angle. By the Pythagorean Theorem, it follows that the hypotenuse of this triangle has length c = va2 + b2, the same as the hypotenuse of the first triangle. Since both triangles’ sides are the same lengths a, b and c, the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle. The above proof of the converse makes use of the Pythagorean Theorem itself.
A corollary of the Pythagorean Theorem’s converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let c be chosen to be the longest of the three sides and a + b ;gt; c (otherwise there is no triangle according to the triangle inequality). The following statements apply. * If a2 + b2 = c2, then the triangle is right. * If a2 + b2 ;gt; c2, then the triangle is acute. * If a2 + b2 ;lt; c2, then the triangle is obtuse.