**How to Find the Distance Between Two Points on a Coordinate Plane ?**

To find the distance between two points in a coordinate plane, we use the formula given below.

d = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

**Example 1 :**

Find the distance between the following pairs of points :

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

**Solution :**

Distance between two points = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Here x_{1} = 2, y_{1} = 3, x_{2} = 4 and y_{2} = 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 = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Here x_{1} = 5, y_{1} = 7, x_{2} = -1 and y_{2} = 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 = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Here x_{1} = a, y_{1} = b, x_{2} = -a and y_{2} = -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 = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Here x_{1} = 0, y_{1} = 0, x_{2} = 36 and y_{2} = 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.

**Example 3 :**

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 x_{1} = 1, y_{1} = 5, x_{2} = 2 and y_{2} = 3

AB = √(2 - 1)² + (3 - 5)²

= √(1)² + (- 2)²

= √1 + 4

= √5

Distance between B and C

Here x_{1} = 2, y_{1} = 3, x_{2} = -2 and y_{2} = -11

BC = √(-2 - 2)^{2} + (-11 - 3)^{2}

= √(-4)² + (-14)²

= √16 + 196

= √212

Distance between C and A

Here x_{1} = -2, y_{1} = -11, x_{2} = 1 and y_{2} = 5

CA = √(1 -(-2))² + (5 -(-11))²

= √(1+2)^{2} + (5 + 11)^{2}

= √3^{2} + 16^{2}

= √9 + 256

= √265

Since AB + BC ≠ CA

After having gone through the stuff given above, we hope that the students would have understood, how to find the distance between two points on a coordinate plane

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