Question

verify whether the points (-2, 3) , (4, 2) and (10, 1) are collinear or not ,​

Answer 1

Given

We have given three points (-2,3),(4,2)&(10,1)

To find

We have to find whether these points are collinear or not

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

Let the points be A(-2,3),B(4,2) &(10,1)

We have to check whether these points are collinear or not

How to check collinearity :

1. Firstly we find the distance between first two points say AB.
2. Then we find distance between another consecutive points say BC & CA
3. Now ,if the sum of distance of any points is equal to the third then it is said to be collinear.

Distance formula

Distance formulaD=√(x₂- x₁)²+(y₂- y₁)²

Distance between A and B

A(-2,3) & B( 4,2)

x₁=-2 ;x₂ = 4 ; y₁= 3 & y₂ = 2

➙ √( 4-(-2))²+(2-3)²

➙√(4+2)²+(-1)²

➙√6²+(-1)²

➙√ 36+1= √37

Hence, AB distance = √37 units.

Distance between B and C

B (4,2) & C( 10,1)

x₁=4 ;x₂ = 10 ; y₁= 2 & y₂ =1

➙BC=√(10-4)²+(1-2)²

➙BC= √(6)²+(-1)²

➙BC= √ 36+1= √37

BC= √37 units

Distance between C and A

C (10,1) & A( -2,3)

x₁=10 ;x₂ = -2 ; y₁= 1 & y₂ = 3

➙CA= √ (10-(-2))²+(1-3)²

➙CA=√(10+2)²+(-2)²

➙CA=√12²+(-2)²

➙CA= √144+4

➙CA=√148= 2√37

CA= 2√37 units .

Here ,we see that AB+BC= CA

➙√37+√37=2√37

Hence,the above all points are collinear.

Answer 2

Let the points be A(-2,3),B(4,2) &(10,1)

We have to check whether these points are collinear or not

How to check collinearity :

Firstly we find the distance between first two points say AB.

Then we find distance between another consecutive points say BC & CA

Now ,if the sum of distance of any points is equal to the third then it is said to be collinear.

Distance formula

Distance formulaD=√(x₂- x₁)²+(y₂- y₁)²

Distance between A and B

A(-2,3) & B( 4,2)

x₁=-2 ;x₂ = 4 ; y₁= 3 & y₂ = 2

➙ √( 4-(-2))²+(2-3)²

➙√(4+2)²+(-1)²

➙√6²+(-1)²

➙√ 36+1= √37

Hence, AB distance = √37 units.

Distance between B and C

B (4,2) & C( 10,1)

x₁=4 ;x₂ = 10 ; y₁= 2 & y₂ =1

➙BC=√(10-4)²+(1-2)²

➙BC= √(6)²+(-1)²

➙BC= √ 36+1= √37

BC= √37 units

Distance between C and A

C (10,1) & A( -2,3)

x₁=10 ;x₂ = -2 ; y₁= 1 & y₂ = 3

➙CA= √ (10-(-2))²+(1-3)²

➙CA=√(10+2)²+(-2)²

➙CA=√12²+(-2)²

➙CA= √144+4

➙CA=√148= 2√37

CA= 2√37 units .

Here ,we see that AB+BC= CA

➙√37+√37=2√37

Hence,the above all points are collinear.