Question

Prove that, the points (1,5),(2,3), and (-2,-11) are not collinear.

Answer 1

Let the points are $A(1,5),B(2,3)$ and $C(-2,-11)$ respectively i.e. co-ordinate of $A$ is $(1,$5), co-ordinate of B is ( 2, 3) and co-ordinate of C is (-2, -11).
$\begin{array}{cc}AB& =\sqrt{(2-1{)}^2+(3-5{)}^2}\\ & =\sqrt{1^2+(-2{)}^2}=\sqrt{1+4}=\sqrt{5}\\ BC& =\sqrt{(-2-2{)}^2+(-11-3{)}^2}\\ & =\sqrt{16+196}\\ & =\sqrt{212}\\ CA& =\sqrt{(1-(-2){)}^2+(5-(-11){)}^2}\\ & =\sqrt{9+256}\\ & =\sqrt{265}\end{array}$
It is clear that sum of distances of any two sides is not equal to third side. So these points are not collinear.