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.
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:
$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$
\(=\sqrt{\left(-3-\left(-1\right)\right)^{2}+\left(5-\left(-2\right)\right)^{2}}\)
$=\sqrt{\left(-3+1\right)^{2}+\left(5+2\right)^{2}}$
$=\sqrt{\left(-2\right)^{2}+\left(7\right)^{2}}$
$\sqrt{4+49}$
$=\sqrt{53}$
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