The Distance Formula

In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. Distance Formula is used to calculate the distance between two points. The distance between (x1, y1) and (x2, y2) is given by:

\[\large d=\sqrt{\left(x_{2}^{2}-x_{1}^{2}\right)+\left(y_{2}^{2}-y_{1}^{2}\right)}\]

Example For The Distance Formula

Question: Given the points (1, -2) and (-3, 5), find the distance between them.

Solution:

Label the points as follows

$\left(x_{1},y_{1}\right)=\left(-1,-2\right) and \left(x_{2},y_{2}\right)=\left(-3,-5\right)$

Therefore: $x_{1}=-1,\: y_{1}=-2,\: x_{2}=-3, and\: y_{2}=5$

To find the distance (d) between the points, use the distance formula:

$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

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

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

$=\sqrt{\left(-2\right)^{2}+\left(7\right)^{2}}$

$\sqrt{4+49}$

$=\sqrt{53}$