What is the Distance Formula?

Hint: Distance formula is used to find the length of the line joining two points in a $x-y$ plane. It is also used to find the distance between two lines also in analytical geometry. It is also used to determine the side of a triangle if only the points of three vertices are given.

Complete step-by-step solution:
Distance formula is used to calculate the length of the line which joins the two points in a $x-y$ plane.
If the two points are given as:
$\left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right)$
Then the formula will be as below:
$d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$
Here, we find the square root of each coordinate point subtracted separately and their square is added.
This formula is derived from Pythagoras Theorem
If the points are in 2d we use above formula but what if the points are in 3d then we have to deal with three coordinate so our formula will be as follows:
If the two points are:
$\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right),\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)$
Then the formula for the above points will be as follows:
$d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}+{{\left( {{z}_{2}}-{{z}_{1}} \right)}^{2}}}$

Note: There are different distance formulas for different kinds of figures like for two parallel lines we have different formulae where we use the line equation and get the distance using the point in it. Then there is a polar coordinate distance formula where we use the Cartesian system to find the distance. The distance between two points can never be negative because in distance negative value has no meaning. Distance formula is very useful to find the distance for many geometrical problems.