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In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance.
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that 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 on the other two sides.
The Pythagorean theorem is used to calculate the distance between points in a plane. Even over short distances, the accuracy of geographic distance calculations which assume a flat Earth depend on the method by which the latitude and longitude coordinates have been projected onto the plane.
It can be found starting with a change of variables that moves the origin to coincide with the given point then finding the point on the shifted plane + + = that is closest to the origin. The resulting point has Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} :
Since it is straightforward to calculate the length of each linear segment (using the Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summation of the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance. [1]
The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
Place two points on two adjacent edges of a unit cube, each at a distance of 3/4 from the point where the two edges meet, and two more points symmetrically on the opposite face of the cube. Then these four points form a square with side length 3 2 4 ≈ 1.0606601. {\displaystyle {\frac {3{\sqrt {2}}}{4}}\approx 1.0606601.}
As a consequence of the Pythagorean theorem, the hypotenuse is the longest side of any right triangle; that is, the hypotenuse is longer than either of the triangle's legs. For example, given the length of the legs a = 5 and b = 12, then the sum of the legs squared is (5 × 5) + (12 × 12) = 169, the square of the hypotenuse.