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A Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation + =; the triple is said to be primitive if and only if the greatest common divisor of a, b, and c is one.
There is a method to construct all Pythagorean triples that contain a given positive integer x as one of the legs of the right-angled triangle associated with the triple. It means finding all right triangles whose sides have integer measures, with one leg predetermined as a given cathetus . [ 13 ]
A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). [1] For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not.
[4] [6] The first three of these define the primitive Pythagorean triples (the ones in which the two sides and hypotenuse have no common factor), derive the standard formula for generating all primitive Pythagorean triples, compute the inradius of Pythagorean triangles, and construct all triangles with sides of length at most 100. [6]
The set of all nodes at a given depth is sometimes called a level of the tree. The root node is at depth zero. Height - Length of the path from the root to the deepest node in the tree. A (rooted) tree with only one node (the root) has a height of zero. In the example diagram, the tree has height of 2. Sibling - Nodes that share the same parent ...
To use modern terminology, if p and q are natural numbers such that p>q then (p 2 − q 2, 2pq, p 2 + q 2) forms a Pythagorean triple. The triple is primitive, that is the three triangle sides have no common factor, if p and q are coprime and not both odd.
English: A depiction of all the primitive Pythagorean triples (a,b,c) with a and b < 1170 and a odd, where a is plotted on the horizontal axis, b on the vertical. The curvilinear grid is composed of curves of constant m − n and of constant m + n in Euclid's formula, a = m 2 − n 2 , b = 2 m n {\displaystyle a=m^{2}-n^{2},b=2mn} .
In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. [1] Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13). A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a ...