Distance Formula | Between Two Points | Point and Line

Distance Formula is used to calculate distance magnitude. This distance can be distance between two points, distance between a point and a line or distance between two parallel lines etc, depending on the formula used. Here we will learn about all the distance formulas used in 2D-Geometry.

What is Distance Formula in 2D?

It is the formula used to measure various distances on the two dimensional plane.

There are different types of distance formulas in 2D geometry.

Distance between two points
Distance from a point to a line
And, Distance between two parallel lines

Distance Between Two Points

Distance formula used to calculate distance between two given points on a two dimensional plane. Also referred as the Euclidean distance formula.

Derivation of the Formula

Let us assume two points A(x₁,x₂) and B(y₁,y₂) on the 2D plane.

We will use Pythagoras theorem to derive the formula.

In the ∆ABC, using pythagoras theorem

We can say,

(AB)² = (BC)² + (AC)²

Hence, d² = (x₂ – x₁)² + (y₂ – y₁)²

Therefore, d = √[((x₂ – x₁)² + (y₂ – y₁)²]

Formula to find Distance Between Two Points

distance formula between 2 points

Distance From a Point To a Line

It is the formula used to find the distance of a given point to a line in the 2D plane. The distance is the length of the perpendicular line segment drawn from the point to the given line.

Consider a line L: Ax + By + C = 0 and a point P(x₁,y₁) on the 2D plane. Then the distance(d) from the point(P) to the line(L) is given as:

Distance formula for point and line

Distance between Two Parallel Lines

It is the distance between two parallel lines is the shortest distance from a point on one line to a point on other line, i.e It is the perpendicular distance between the two parallel lines.

Let the two lines be L₁: Ax₁ + By₁ + C₁ = 0 and L₂: Ax₂ + By₂ + C₂ = 0, let the perpendicular distance between the two lines be d.

Then the formula for Distance between Two Parallel Lines is given as:

Illustrative Examples on Distance Formula

Example 1: Find the distance between the points (-3, 2), and (4, -3).

Solution: The given two points are (x₁,y₁) = (-3, 2), and (x₂,y₂) = (4, -3)

Using the Euclidean distance formula,

Distance = √[(x₂−x₁)²+(y₂−y₁)²]

= √[(4+3)²+(-3-2)²]

? √[7²+(-5)²] = √(49+25)

= √74

Answer: Distance = √74

Example 2: Find the distance from the point (2, -3) to the line 3x – 4y = 2.

Solution: The given point is, (x₁,y₁)= (2, -3).

We can write line equation as 3x – 4y – 2 = 0. Comparing this with Ax + By + C = 0, we get A = 3, B = -4, and C = -2.

Using the distance formula to find the distance from a point to a line,

d = |(3×2)+(-4×-3)+(-2)|/√[3²+(-4)²]

=> d = |6+12-2|/√(9+16)

Hence, d = 16/5

Answer: Distance = 16/5

Picked For You

Furthermore, these are some topics which might interest you:

To solve and ask more and more questions download Kunduz app for free.