Distance Formula

The distance between two points is measured as the length of the interval which joins two points. If the two points lie on the same horizontal or same vertical line then the distance is being calculated by subtracting the coordinates which are different. In the analytical geometry, the distance formula is useful to compute the distance that measures between two lines also. It is also useful to find the sum of the lengths of all the sides of a polygon, an area of polygons, and many more. It is also useful to determine a given triangle as scalene, isosceles, or equilateral.

How to Find the Distance Between Two Points

The distance between the two points along the XY-plane can be found using the distance formula. An ordered pair (x, y) represents the coordinate of the point, where x-coordinate (or abscissa) is the distance of the point from the centre and y-coordinate (or ordinate) is the distance of the point from the centre.

The Distance Formula is a useful method to find the length between two points in any shape. We may represent these points using coordinate geometry. This formula itself is actually derived with the help of the Pythagorean Theorem.

This theorem establishes a relationship as bellow,

c2 = a2 + b2

Where c is the longest side of a right triangle means hypotenuse. And a and b are the other shorter sides means perpendicular and base. The main purpose of the Distance Formula is to compute the length of the hypotenuse of the right triangle represented as c.

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Explanation of the Distance Formula Between Two Points in a 2d Plane:

This derivation is based on the Pythagoras theorem. Let us consider two points P (x1, y1) and Q (x2, y2) on the given coordinate axis. Then the distance between these points will be as given:

D = \[\sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}\]

The expression x2 - x1 is referred to as the difference of the x-coordinate values of both points. Also, the expression y2 - y1 referred to as the difference of the y-coordinate values of both points.

Explanation of The Distance Formula Between Two Points in a 3d Plane

AS in the 2d plane, now let us consider two points P (x1, y1, z1)and Q (x2, y2, z2)on the given coordinate axis of the 3d plane. Then the distance between these points will be as given:

D = \[\sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{^{2}}}\]

Distance Formula of a Point Form a Given Line: 

It is the perpendicular distance of a point from a line. We can measure the distance of the point from a given line.

Let us consider the equation of a straight line is Ax + By + C = 0, where A, B, and C are the coefficients. Also, assume that the point P in the Cartesian coordinate system with coordinates (x1, y1). Then the distance between line and point will be the length of the perpendicular drawn from point P to the line.  

The distance formula from a point to a line is as given below:

d = \[\frac{|Ax_{1} + By_{1} + C|}{\sqrt{A^{2} + B^{2}}}\]

Minimum Distance Between Two Parallel Lines

The distance between parallel lines is the minimum distance from any point on one of the lines to the other line. This distance is defined as the distance between two parallel lines, and obviously, it will be the perpendicular distance between them.

Let us assume the two straight lines parallel to each other as:

The first line is,

Line-1 : y = mx + c1 , this equation is in slope-intercept form.

And the second line is

Line-2 : y = mx + c2 , this equation is in slope-intercept form.

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Let us consider that ‘d’ is the distance between both the lines.

Formula to find the minimum distance between two non-intersecting lines as given below:

d = \[\frac{|C_{1} - C_{2}|}{\sqrt{A^{2} + B^{2}}}\]

Distance Between Two Points in Polar Coordinates: 

We have used the Cartesian system for finding out the distance. Now, we will use the polar coordinate system for the same purpose. We know that in the polar coordinate system, the same point will be having polar coordinates in terms of r and

So, let us assume that O be the pole and OX be the initial line. Let P and Q be two given points whose polar coordinates are ( \[r_{1}, \theta_{1}\] ), and ( \[r_{2}, \theta_{2}\] ) respectively.  Now, the distance formula between the points is given by,

d = \[\sqrt{r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{1}- \theta_{2})}\]

Also, here \[\theta_{1}, \theta_{2}\] are the angles measured in radians.

Fun Facts:

• It is a scalar quantity, as it does not depend on the direction. 

• Distance between two points depends on the coordinates of the endpoints.

• Distance between two points cannot be negative. Because negative value has no meaning.

• Distance between two points helps to prove many geometrical problems and principles as well.

• The distance formula is also applicable in polar coordinate systems.