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The spiral is started with an isosceles right triangle, with each leg having unit length.Another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3.
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.
A Pythagorean triple has three positive integers a, b, and c, such that a 2 + b 2 = c 2. 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).
An example of "beauty in method"—a simple and elegant visual descriptor of the Pythagorean theorem.. Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics.
The Pythagorean prime 5 and its square root are both hypotenuses of right triangles with integer legs. The formulas show how to transform any right triangle with integer legs into another right triangle with integer legs whose hypotenuse is the square of the first triangle's hypotenuse.
Early-Pythagorean philosophers proved simple geometrical theorems, including "the sum of the angles of a triangle equals two right angles". Pythagoreans also came up with three of the five platonic solids : the tetrahedron , the cube and the dodecahedron .
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem [1] or the upside down Pythagorean theorem [2]) is as follows: [3] Let A, B be the endpoints of the hypotenuse of a right triangle ABC. Let D be the foot of a perpendicular dropped from C, the vertex of the right angle, to the hypotenuse. Then
The tetractys. The tetractys (Greek: τετρακτύς), or tetrad, [1] or the tetractys of the decad [2] is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number.