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)+(-5+3))^(2))`
` =sqrt( 1+(-2)^(2))`
` = sqrt(1+4)`
= sqrt(5) `…(i)
`d(B,C=)= sqrt((-4) -2)^(2) + [7-(-5)]^(2))`
`= sqrt((-6)^(2) + (12)^(2))`
` = sqrt(38 + 144)`
` = sqrt(180)`
` sqrt(2xx2xx3xx3xx5)`
` = 6 sqrt(5)` ...(2)
`d(A,C) = sqrt((-4-1)^(2) + [7-(-3)]^(2))`
` = sqrt((-5)^(2) + (10)^(2))`
`sqrt( 25 + 100) = sqrt(125)`
`sqrt(5xx5xx5xx5)`
` = 5sqrt(5)` ...(3)
Adding (1) and (3)
`d (A , B) + d(A,C)= sqrt(5) + 5sqrt(5) = 6 sqrt(5)`
` therefore A(A,B + d(A,C) = d (B,C)" "` ...From (2)] ...(4)
`therefore ` Point A,B and C are collinear.
Point A(1,-3) , (2,-5) and C(-4,7) are colliear.