How are the distance formula and pythagorean theorem related?

Answer 1

The distance formula IS the Pythagorean theorem, applied to a right triangle with the x-coordinate and y-coordinate as the two shorter sides. Or the equivalent in 3, 4, or more dimensions in flat (i.e., Euclidian) space.

Answer 2

There are many different ways of measuring distance between two points and one of these is the Pythagorean distance. Another measure, for example, is the Minkowski metric (also known as the taxicab distance). This is based on the grid-like layour of Manhattan and is the sum of the number of units (blocks) in the north / south directions added to the number of units in the east / west directions.

From its very name, it is clear that the Pythagorean distance is based on the Pythagoras theorem. In 2-dimensional space, the distance is the square root of the sum of the squares of the distances in two mutually perpendicular directions.

The mutually perpendicular directions are normally the x and y axes in the coordinate plane, and the distance between A = (p, q) and B = (r, s) is

sqrt[(p - r)^2 + (q - s)^2].

Consider the point C = (r, q)

Then, in triangle PQR, |p - r|, which is the difference in abscissae (x coordinates) of A and B which is also the horizontal distance between A and C.

Similarly, |q - s|, which is the difference in ordinates of A and B, and this is also the vertical distance between B and C.

Finally, since the axes are mutually perpendicular, angle C is a right angle.

So, by Pythagoras, AC^2 + BC^2 = (p - r)^2 + (q - s)^2 = AB^2

Taking square roots, gives the distance formula.

The result can be generalised to more dimensions. For example, in 3-d,

if A = (u, v, w) and B = x, y, z) then

AB= sqrt[(u - x)^2 + (v - y)^2 + (w - z)^2].