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Distance Between Two Points

Last Updated : 28 Apr, 2024
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“Distance Between Two Points” is a fundamental topic in the branch of mathematics known as Coordinate Geometry. Coordinate geometry is a branch of mathematics that discusses the relationship between geometry and algebra using graphs comprising curves and lines. Allowing some geometric aspects which are provided in Algebra in the Cartesian plane helps students to solve various geometric problems which can’t be solved by only concepts of one branch of mathematics.

In coordinate geometry, the coordinates of points on a plane are expressed as an ordered pair of numbers and we can also perform operations like finding the distance between two points, dividing lines in m:n ratios, identifying the mid-point of a line, calculating the area of a triangle in the cartesian plane, and so on.  In this article, we will learn how to find the distance between two points. Let’s first understand what are point before learning the methods and formula to find the distance between them.

Table of Content

What are Points in Cartesian Plane?

In mathematics, a location in any space or a cartesian plane is known as a point with no size, no height, no length, or not even shape. It is generally used to mark the beginning of a shape or a diagram. The point is represented as shown in the below image:

Points in Cartesian Plane

And the straight path that connects these points are known as a line. The types of points are:

Collinear and Non-Collinear Points

When 3 or more points are present on the straight line then such types of points as known as Collinear points and if these points do not present on the same line then such types of points are known as non-collinear points. 

Collinear and Non-Collinear Points

Coplanar and Non-Coplanar Points

When the group of points is present on the same plane then such types of points are known as coplanar points and if these points do not present on the same plane then such types of points are known as non-coplanar points.

Coplanar and Non-Coplanar Points

How to Find Distance Between Two Points?

In general, distance is used to find how far two objects are separated. Similarly in mathematics, distance is used to find the distance between two points according to their given coordinates. It should be noted that the points in question do not necessarily have to be in the same quadrant. The distance formula is generally used in the coordinate system in mathematics to calculate how far two points are present in a coordinate plane which makes it a very important topic in coordinate geometry.

For the sake of simplicity, 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:

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

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

= (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.

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 = \sqrt{{(5-2)}^2+{(10-4)}^2}

⇒ D = \sqrt{{(3)}^2+{(6)}^2}

⇒ D = \sqrt{9+36}

⇒ D = \sqrt{9+36}

⇒ D = \sqrt{45} = 3\sqrt{5}

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

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 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:

\bold{Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}

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 = \sqrt{{(3-(-1))}^2+{(-2-5)}^2+{(4-2)}^2}

⇒ D = \sqrt{{(4)}^2+{(-7)}^2+{(2)}^2}

⇒ D =  \sqrt{16+49+4} = \sqrt{69}

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

Read More,

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 = \sqrt{{(a-p)}^2+{(b-q)}^2}

Now put the value in the formula

⇒ AB = \sqrt{{(4-1)}^2+{(6-0)}^2}

= \sqrt{3^2 + 6^2}

= \sqrt{45}  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 = \sqrt{{(a-p)}^2+{(b-q)}^2}

Now put the value in the formula

⇒ PQ = \sqrt{{(4-1)}^2+{(0-0)}^2}

= \sqrt{3^2 + 0^2}

= \sqrt{9}  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 \bold{Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}

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

⇒ AB= \sqrt{2^2 + 0^2 + 1^2}

⇒ AB= \sqrt{5}  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 = \sqrt{{(a-p)}^2+{(b-q)}^2}

Now put the value in the formula

⇒ PR = \sqrt{{(6-4)}^2+{(0-0)}^2}

= \sqrt{2^2 + 0^2}

= 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 = \sqrt{{(a-p)}^2+{(b-q)}^2}

Now put the value in the formula

⇒ PR = \sqrt{{(12-4)}^2+{(0-0)}^2}

= \sqrt{8^2 + 0^2}

= 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 = \sqrt{{(a-p)}^2+{(b-q)}^2}

Now put the value in the formula

⇒ AB = \sqrt{{(12-10)}^2+{(0-0)}^2}

= \sqrt{2^2 + 0^2}

= 2 units

FAQs on Distance Between Two Points

What is Distance Between Two Points?

The distance between two points is nothing but the length of the straight line segement joining those points i.e., it is the shortest distance between the two points.

How to Find Distance Between Two Points in 2D?

We can find the distance between two points  (x1, y1) and (x2, y2) using the distance formula as follows:

\bold{\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}

How to Find Distance Between Two Points in 3D?

For two points with three-dimensional coordinates (x1,y1,z1) and (x2,y2,z2), the distance between them is given by as follows:

\bold{Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}

Can you Find Distance Between Two Points in a Coordinate Plane without using the Distance Formula?

Yes, we can also find the distance between two points in a coordinate plane by drawing a right angle triangle using both points as end of hypotenous and applying  Pythagorean theorem to find the length of hypotenuse.



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