# How do you find the distance between (3,-2) and (5,-3)?

Aug 27, 2016

Use the distance formula to find the distance is $\sqrt{5}$.

#### Explanation:

The (Euclidean) distance $d$ between points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ is given by the distance formula:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

In our example, $\left({x}_{1} , {y}_{1}\right) = \left(3 , - 2\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(5 , - 3\right)$, so we have:

$d = \sqrt{{\left(5 - 3\right)}^{2} + {\left(- 3 - \left(- 2\right)\right)}^{2}} = \sqrt{{2}^{2} + {\left(- 1\right)}^{2}} = \sqrt{4 + 1} = \sqrt{5}$

Aug 27, 2016

See below.

#### Explanation:

Suppose you are in Manhattan Center and you need to walk from 399 West 14th street to Empire State Building. What is the shortest feasible path to perform this walk? How do you measure distances in this case? Depending on the terrain topology there are more adequate ways of measuring distances. For the afore mentioned Manhattan trip, a feasible distance formula between points ${p}_{1} = \left({x}_{1} , {y}_{1}\right)$ and ${p}_{2} \left({x}_{2} , {y}_{2}\right)$ could be

$d \left({p}_{1} , {p}_{2}\right) = \left\mid {x}_{1} - {x}_{2} \right\mid + \left\mid {y}_{1} - {y}_{2} \right\mid = \left\mid 3 - 5 \right\mid + \left\mid - 2 + 3 \right\mid = 3$

There are a lot of ways of distance measure. What criteria must obey a measure to be a distance? 