Using distance formula decide whether the points (4 3), (5 1) and (1 9) are collinear or not

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Let A(x1, y1) = A(4, 3), B(x2, y2) = B(5, 1), C(x3, y3) = C(1, 9)

∴ d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

= `sqrt((5 -4)^2 + (1 - 3)^2`

= `sqrt(1^2 + (-2)^2`

= `sqrt(1+ 4)`

= `sqrt(5)`   ......(i)

∴ d(B, C) = `sqrt((x_3 - x_2)^2 + (y_3 - y_2)^2`

= `sqrt((1 - 5)^2 + (9 - 1)^2`

= `sqrt((-4)^2 + 8^2`

= `sqrt(16 + 64)`

=`sqrt(80)`

= `4sqrt(5)` ......(ii)

∴ d(A, C) = `sqrt((x_3 -x_1)^2 + (y_3 -  y_2)^2`

= `sqrt((1 -  4)^2 + (9  - 3)^2`

= `sqrt((-3)^2 + 6^2`

= `sqrt(9 + 36)`

= `sqrt(45)`

= `3sqrt(5)`  .....(iii)

`sqrt(5) + 3sqrt(5) = 4sqrt(5)`

∴ d(A, B) + d(A, C) = d(B, C)   ......[From (i), (ii) and (iii)]

∴ Points A, B, C are collinear.