Distance Formula

Learn about distance formula here and understand the concept better with solved examples provided. Students can also use the online distance formula calculator and distance formula worksheet provided in the page.

Let's understand what is the distance formula? The length of a line segment AB, which joins A (x₁, y₁) and B (x₂, y₂) is given by,

Let A (x₁, y₁) and B (x₂, y₂) be two points in the plane.

Let d = distance between the points A and B.

Draw AL and BM perpendicular to x-axis (parallel to y-axis).

Draw AC perpendicular to BM to cut BM at C.

In the figure,

OL = x₁, OM = x₂ [AC = LM = OM - OL = x₂ - x₁]

MB = y₂, MC = LA = y₁ [CB = MB - MC = y₂ - y₁]

From the right-angled DACB,

Note:

i) If the points A and B lie on the x-axis, then the ordinates of A and B are zeros.

i.e., A (x₁, 0), B (x₂,0)

ii) If the points A and B lie on the y-axis, then the abscissae of A and B are zeros.

i.e., A (0,y₁) and B (0,y₂)

iii) Distance of any point A (x, y) from the origin

Below are some examples based on distance formula

Example 1: Find the distance between the following pair of points: A (1,2) and B (4,5).

Solution:

Using the distance formula, we have

Example 2: Find the distance between places when the two coordinates (2, 4) and (4, 6)are given, using the distance formula.?

Solution:

(x₁, y₁)= (2, 4)

(x₂, y₂) = (4, 6)

Distance= `sqrt((y2-y1)^2 + (x2-x1)^2)`

Here (x₁, y₁) and (x₂, y₂) are two places. We need to find the distance the two places

Distance=`sqrt((4-2)^2 + (6 -4)^2)`

Distance=`sqrt((2)^2 + (2)^2)`

Distance= `sqrt(8)`

Distance= 2.82 units

Example 3: Find the distance between places when the two coordinates (10, 15) and (15, 20)are given, using the distance formula.?

Solution:

(x₁, y₁)= (10, 15)

(x₂, y₂) = (15, 20)

Distance=`sqrt((y2-y1)^2 +(x2-x1)^2) `

Here (x₁, y₁) and (x₂, y₂) are two places. We need to find the distance the two places

Distance=`sqrt((15-10)^2 + (20-15 )^2)`

Distance=`sqrt((5)^2 + (5)^2)`

Distance= `sqrt(50)`

Distance = 7.07 units