# 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)

=  √(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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