Determine whether the points A(1, -3) , B(2,-5) and
C(-4,7) are collinear or not .

Explanation

`A(1,-3),B(2,-5)` and `C(-4,7)`
By distance formula
`d(A,B)=sqrt((2-1)^(2)+[-5-(-3)]^(2))`
`=sqrt((1)^(2)+(-5+3)^(2))`
`=sqrt(1+(-2)^(2))`
`=sqrt(1+(-2)^(2))`
`=sqrt(1+4)`

`=sqrt(5)`……..1
`d(B,C)=sqrt((-4-2)^(2)+[7-(-5)^(2)]^(2))`
`=sqrt((-6)^(2)+(12)^(2))`
`=sqrt(180)`
`=sqrt(2xx2xx3xx3xx5)`
`=6sqrt(5)` ...........2
`d(A,C)=sqrt((-4-1)^(2)+[7-(-3)]^(2))`
`=sqrt((-5)^(2)+(10)^(2))`
`=sqrt(25+100)=sqrt(125)`
`=sqrt(5xx5xx5)`
`=5sqrt(5)`........3
ADding 1 and 3
`d(A,B)+d(A,C)=sqrt(5)+5sqrt(5)=6sqrt(5)`........4
`:.d(A,B)+(A,C)=d(B,C)`...[Form 2 and 4]
`:.` points A,B and C are collinear.
Points `A(1,-3),B(2,-5)` and `C(-4,7)` are collinear.