# Finding the distance between 2 points

Given any 2 points $(x_1,y_1)$ and $(x_2,y_2)$ , finding the distance between 2 points involves the following 3 simple steps:

1. State the Distance formula
2. Substitute the given points $(x_1,y_1)$ and $(x_2,y_2)$ into the Distance formula
3. Simplify to find the final solution

## The Distance Formula

The formula for finding the distance between 2 points $(x_1,y_1)$ and $(x_2,y_2)$ is given by: $$D = \sqrt{(x_2-x_1)^2+ (y_2-y_1)^2}$$ The graphics above shows that we can denote $x_1 = 3$ and $y_1 =2$ for the point (3,2) and $x_2=12$ and $y_2 = 8$ for the point (12,8)

### Worked example

Find the distance of the line joining the points (3,2) and (12,8) shown in the diagram above

For solving the above problem, we begin by stating the distance formula followed by substitution and simplification as follows:
\begin{align} D& = \sqrt{(x_2-x_1)^2+ (y_2-y_1)^2} & &\color{green}{\text{state the distance formula}}\\ & = \sqrt{(12-3)^2+ (8-2)^2} & &\color{green}{\text{substitution : $x_1=3,y_1=2,x_2=12,y_2=8$}}\\ & = \sqrt{9^2+ 6^2} & &\color{green}{\text{Simplify bracket terms 12-3=9 and 8-2=6}} \\ & = \sqrt{81+ 36} & &\color{green}{\text{Find the squares $9^2=81$ and $6^2=36$}}\\ & = \sqrt{117} & &\color{green}{\text{Add the squares: 81+36=117}} \\ & = \sqrt{9 \times 13} & &\color{green}{\text{The square root is not an integer, so we may express in surd form}} \\ & = \sqrt{9}\sqrt{13} \\ & = 3\sqrt{13} \end{align}

## Distance between 2 points Calculator

You can visit Distance between 2 points Calculator here . The distance calculator provides detailed solution for distance when 2 coordinates are entered. It can be used as a homework helper.