$d = \sqrt{{(x_{2}-x_{1})}^2+{(y_{2}-y_{1})}^2}$
It is a distance formula and used to find the distance between any two points in a two dimensional Cartesian coordinate system. Now, learn how to derive the distance formula in geometry.
$\overline{PQ}$, $\overline{QR}$ and $\overline{PR}$ are hypotenuse, opposite side (perpendicular) and adjacent side (Base) of right triangle $RPQ$. Now, calculate the length of each side in terms of coordinates of the points.
Use this data to find the distance between any two points in a two dimensional Cartesian coordinate system.
The relation between three sides can be written in mathematical form by Pythagorean Theorem.
${PQ}^2 = {PR}^2+{QR}^2$
Substitute lengths of the all three sides.
$\implies d^2 = {(x_2-x_1)}^2+{(y_2-y_1)}^2$
$\implies d = \pm \sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
The distance is a positive factor physically.
$\,\,\, \therefore \,\,\,\,\,\, d = \sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
It is called distance formula and used to find distance between any points in a plane. The distance formula reveals that the distance between any two points in a plane is equal to square root of sum of squares of differences of the coordinates.
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