HOW TO FIND THE DISTANCE  BETWEEN TWO POINTS ON A COORDINATE PLANE

The formula to find the distance between the two points (x₁, y₁) and (x₂, y₂) is given by

d  =  √(x₂ - x₁)² + (y₂ - y₁)²

Example 1 :

Find the distance between the following pairs of points :

(i) (2 , 3) (4 , 1)

Solution :

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = 2, y₁ = 3, x₂ = 4  and  y₂ = 1

  =  √(4 - 2)² + (1 - 3)²

   =  √2² + (-2)²

  =  √(4 + 4)

  =  √8

  =  2 √2

(ii)  (-5 , 7) (-1 , 3)

Solution :

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = 5, y₁ = 7, x₂ = -1  and  y₂ = 3

  =  √(-1 -(-5))² + (3 - 7)²

  =  √(-1 + 5)² + (-4)²

  =  √4² + (-4)²

  =  √16 + 16

  =  √32

  =  4 √2

(iii)  (a , b) (-a , -b)

Solution :

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = a, y₁ = b, x₂ = -a  and  y₂ = -b

  =  √(-a -a)² + (-b - b)²

  =  √(-2a)² + (-2b)²

  =  √4a² + 4b²

  =  √4(a² + b²)

  =  √2 x 2(a² + b²)

  =  2 √(a² + b²)

Example 2 :

Find the distance between the points (0, 0) and (36, 15). 

Solution :

Let A (0,0) B (36,15)

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁  =  0, y₁  =  0, x₂  =  36  and  y₂  =  15

  =  √(36 - 0)² + (15 - 0)²

  =  √(36)² + (15)²

  =  √1296 + 225

  =  √1521

  =  √39 x 39

  =  39

So,  the distance between the given towns A and B will be 39 km.

How to check if the given points are collinear using distance formula ?

Example :

Determine if the points (1,5) (2,3) and (-2,-11) are collinear.

Solution :

A (1, 5)  B(2, 3) and C (-2, -11)

Distance between A and B

Here x₁  =  1, y₁  =  5, x₂  =  2  and  y₂  =  3

AB  =  √(2 - 1)² + (3 - 5)²

  =  √(1)² + (- 2)² 

  =  √1 + 4

  =  √5

Distance between B and C

Here x₁  =  2, y₁  =  3, x₂  =  -2  and  y₂  =  -11

  BC  =  √(-2 - 2)² + (-11 - 3)²

  =  √(-4)² + (-14)² 

  =  √16 + 196

  =  √212

Distance between C and A

Here x₁  =  -2, y₁  =  -11, x₂  =  1  and  y₂  =  5

CA  =  √(1 -(-2))² + (5 -(-11))²

  =  √(1+2)² + (5 + 11)²

  =  √3² + 16²

  =  √9 + 256

  =  √265

Since AB + BC ≠ CA

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