Distance Between Two Points

Last Updated : 03 Sep, 2024

Distance Between Two Points is the length of line segment that connects any two points in a coordinate plane in coordinate geometry. It can be calculated using a distance formula for 2D or 3D. It represents the shortest path between two locations in a given space.

In this article, we will learn how to find the distance between two points formula and calculation. Let’s first understand what are point before learning the methods and formula to find the distance between them.

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Table of Content

Distance Formula in a 2D Plane
Distance Formula in a 3D Plane
Sample Problems on Distance Between Two Points

What is the Distance Between Two Points?

The distance between two points in a plane or space is the length of the straight line segment that connects them. This distance can be calculated using the distance formula, which is derived from the Pythagorean theorem

In this article, we will discuss formula and methods to find the distance between points in 2D and 3D coordinate plane.

Distance Formula for Two Points in a 2D Plane

Assume there are two points, A and B, in a coordinate plane, the first quadrant. (a, b) are the coordinates of point A, and (a, b) are the coordinates of point B. (p, q). The distance between points A and B abbreviated AB, must be calculated as follows:

[Tex]\bold{\text{AB} = \sqrt{{(a-p)}^2+{(b-q)}^2}} [/Tex]

How to Find Distance Between Two Points?

To calculate the distance between two points, we can use the following steps:

Identify the coordinates of the two points.
Use the formula discussed above to calculate distance between two point in two dimensional plane.

Let’s discuss an example for the same.

Example: Find the distance between two points X(5, 10) and Y(2, 4).

Solution:

As given points are X(5, 10) and Y(2, 4).

So, the distance between them is using the formula is

D = [Tex]\sqrt{{(5-2)}^2+{(10-4)}^2} [/Tex]

⇒ D = [Tex]\sqrt{{(3)}^2+{(6)}^2} [/Tex]

⇒ D = [Tex]\sqrt{9+36} [/Tex]

⇒ D = [Tex]\sqrt{9+36} [/Tex]

⇒ D = [Tex]\sqrt{45} = 3\sqrt{5} [/Tex]

So, the distance between X and Y is 3√5 units.

Derivation for Distance Formula

Let us assume that two points are present on a 2-dimensional plane that is A and B with coordinates (a, b) and (p, q). Now we construct a right angle triangle i.e.AJB in which AB is a hypotenuse. Now we find the distance between points A and B.

By Pythagoras Theorem,

AB² = AJ² + BJ²

⇒ AB² = (a – p)² + (b – q)²

⇒ AB = √{(a – p)² + (b – q)²}

Let’s consider an example to learn how to use the formula.

Distance Formula in a 3D Plane

Let’s consider the two points in three dimensions to be (x₁, y₁, z₁) and (x₂, y₂, z₂). Thus, the distance between them is given by as follows:

[Tex]\bold{Distance = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}} [/Tex]

How to find the Distance Between Two Points in 3D?

If the given points have coordinates in three dimensions i.e., (x, y, z) then the distance between them can be calculated using the formula given for three dimensions. Let’s consider an example for the same.

Example: Find the distance between two points A(3, -2, 4) and B(-1, 5, 2).

Solution:

Given points are A(3, -2, 4) and B(-1, 5, 2).

So, the distance between them is using the formula is

D = [Tex]\sqrt{{(3-(-1))}^2+{(-2-5)}^2+{(4-2)}^2} [/Tex]

⇒ D =[Tex] \sqrt{{(4)}^2+{(-7)}^2+{(2)}^2} [/Tex]

⇒ D = [Tex]\sqrt{16+49+4} = \sqrt{69} [/Tex]

So, the distance between A and B is √69 units.

Read More,

Coordinate Geometry
Mid-point Formula
Section Formula

Sample Problems on Distance Between Two Points

Problem 1: Find the distance between points A (4, 6) and B(1, 0).

Solution:

Given: A(4, 6) and B(1, 0).

Now we find the distance between the given points that is A and B

So we use the formula

D = [Tex]\sqrt{{(a-p)}^2+{(b-q)}^2} [/Tex]

Now put the value in the formula

⇒ AB = [Tex]\sqrt{{(4-1)}^2+{(6-0)}^2} [/Tex]

= [Tex]\sqrt{3^2 + 6^2} [/Tex]

= [Tex]\sqrt{45}[/Tex] units

= 3√5 units

Problem 2: Find the distance between points P(4, 0) and Q(1, 0).

Solution:

Given: P(4, 0) and Q(1, 0).

Now we find the distance between the given points that is P and Q

So we use the formula

D = [Tex]\sqrt{{(a-p)}^2+{(b-q)}^2} [/Tex]

Now put the value in the formula

⇒ PQ = [Tex]\sqrt{{(4-1)}^2+{(0-0)}^2} [/Tex]

= [Tex]\sqrt{3^2 + 0^2} [/Tex]

= [Tex]\sqrt{9} [/Tex] units

= 3 units

Problem 3: Given points A(3, 0, 4) and B(1, 0, 3). Find the distance between them.

Solution:

Given: A(3, 0, 4) and B(1, 0, 3).

Now we find the distance between the given points that is A and B

Using formula [Tex]\bold{Distance = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}} [/Tex]

⇒ AB = [Tex]\sqrt{{(3-1)}^2+{(0-0)}^2+{(4-3)}^2} [/Tex]

⇒ AB= [Tex]\sqrt{2^2 + 0^2 + 1^2} [/Tex]

⇒ AB= [Tex]\sqrt{5} [/Tex] units

Thus, distance between A and B is √5 units.

Problem 4: Given points P(6, 0) and R(4, 0). Find the distance between them.

Solution:

Given: P(6, 0) and R(4, 0).

Now we find the distance between the given points that is P and R

So we use the formula

D = [Tex]\sqrt{{(a-p)}^2+{(b-q)}^2} [/Tex]

Now put the value in the formula

⇒ PR = [Tex]\sqrt{{(6-4)}^2+{(0-0)}^2} [/Tex]

= [Tex]\sqrt{2^2 + 0^2} [/Tex]

= 2 units

Problem 5: Find the distance between the points (12, 0) and (4, 0).

Solution:

Given: P(12, 0) and R(4, 0).

Now we find the distance between the given points that is P and R

So we use the formula

D = [Tex]\sqrt{{(a-p)}^2+{(b-q)}^2} [/Tex]

Now put the value in the formula

⇒ PR = [Tex]\sqrt{{(12-4)}^2+{(0-0)}^2} [/Tex]

= [Tex]\sqrt{8^2 + 0^2} [/Tex]

= 8 units

Problem 6: Find the distance between the points (12, 0) and (10, 0).

Solution:

Given: A(12, 0) and B(10, 0).

Now we find the distance between the given points that is A and B

So we use the formula

D = [Tex]\sqrt{{(a-p)}^2+{(b-q)}^2} [/Tex]

Now put the value in the formula

⇒ AB = [Tex]\sqrt{{(12-10)}^2+{(0-0)}^2} [/Tex]

= [Tex]\sqrt{2^2 + 0^2} [/Tex]

= 2 units