Math Doubts

Geometric Proof of Distance formula


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

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.

Construct a figure to derive distance formula

distance between points
  1. $P{(x_{1}, y_{1})}$ and $Q{(x_{2}, y_{2})}$ are two points in two dimensional space.
  2. 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.
  3. 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 $\Delta RPQ$.

Calculate lengths of the sides of triangle

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

right triangle to find distance between points
  1. Length of Opposite side is $QR$ $\,=\,$ $OQ-OR \,=\, y_2-y_1$.
  2. Length of Adjacent side is $PR$ $\,=\,$ $OR-OP \,=\, x_2-x_1$.
  3. 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.

Express relation between sides of triangle

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.

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more