Distance formula derivation
The distance formula comes directly from the Pythagorean theorem. The horizontal change and vertical change form the legs of a right triangle, and the distance between the points is the hypotenuse.
Euclidean vs Manhattan distance
Euclidean distance is the straight-line distance. Manhattan distance adds the horizontal and vertical movement, like travelling through city blocks.
Distance in coordinate geometry
Coordinate distance is used in geometry, physics, graphics, mapping, engineering, statistics, and machine learning whenever positions need to be compared.