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

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.

- $P{(x_{1}, y_{1})}$ and $Q{(x_{2}, y_{2})}$ are two points in two dimensional space.
- Join the points by a line and it forms a line segment $\small \overline{PQ}$. The length of line segment is equal to the distance between the points $P$ and $Q$ geometrically.
- Draw a parallel line from point $P$ and a perpendicular line from $Q$ towards $x$-axis. The two lines get intersected at point $R$ perpendicularly and form a right triangle, known as $\Delta RPQ$.

$\overline{PQ}$, $\overline{QR}$ 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.

- Length of Opposite side is $QR$ $\,=\,$ $OQ-OR \,=\, y_2-y_1$.
- Length of Adjacent side is $PR$ $\,=\,$ $OR-OP \,=\, x_2-x_1$.
- Length of Hypotenuse is considered as $d$ and it represents the distance between two points. Therefore, $PQ = d$

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.

Latest Math Topics

Dec 13, 2023

Jul 20, 2023

Jun 26, 2023

Latest Math Problems

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved