# Distance and Midpoint Formulas

## Introduction

The Distance and midpoint formula allow us to talk about the simplest interesting geometric objects, lines. There is a saying that the shortest path between two points is a line, but without a way to measure distance we are unable to determine the shortest distance. The midpoint of a line gives some insight into more geometric concepts.

## Distance Formula

Given two points P and Q with coordinates ${\displaystyle (x_{1},y_{1}){\text{ and }}(x_{2},y_{2})}$, respectively. One way to think about the distance between P and Q is to draw the line segment between P and Q, and find a right triangle where the line segment PQ is the hypotenuse.

Example: Find the distance between (1, 3) and (5, 6).

Solution: In order to form the right triangle we notice that to get from (1, 3) to (5, 6) we could travel 4 units to the right then 3 units up. So you could travel from (1, 3) to (5, 3) then finally to (5, 6). Thus, we have created a triangle with vertices (1, 3), (5, 6), and (5, 3). We also have the additional property that the hypotenuse is the segment between (1, 3) and (5, 6). The side lengths of this triangle are 3, 4 and some unknown value for the hypotenuse. Thus, the distance from (1, 3) to (5, 6) is ${\displaystyle {\sqrt {3^{2}+4^{2}}}={\sqrt {9+16}}=5.}$

## Midpoint Formula

The midpoint between two points P and Q is the point on the line segment PQ that is halfway between P and Q. The formula for the midpoint is ${\displaystyle \left({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}}\right)}$, where the coordinates of P are ${\displaystyle (x_{1},y_{1})}$ and the coordinates of Q are ${\displaystyle (x_{2},y_{2})}$

Example:

Find the midpoint of the line segment between P(-1, 5) and Q( 4, 3)

Solution. Using the formula the midpoint is ${\displaystyle \left({\frac {-1+4}{2}},{\frac {5+3}{2}}\right)=\left({\frac {3}{2}},4\right)}$