Question

Are the three points $A(2,3),B(5,6)$ and $C(0,2)$ collinear?

Answer 1

Three points $A$, $B$ and $C$ are collinear if $AB+BC=AC$.

Therefore, we calculate:

$AB=\sqrt{{(5-2)}^2+{(6-3)}^2}=\sqrt{3^2+3^2}=\sqrt{9+9}=\sqrt{18}$

$BC=\sqrt{{(0-5)}^2+{(2-6)}^2}=\sqrt{(-5{)}^2+(-4{)}^2}=\sqrt{25+16}=\sqrt{41}$

$AC=\sqrt{{(0-2)}^2+{(2-3)}^2}=\sqrt{(-2{)}^2+(-1{)}^2}=\sqrt{4+1}=\sqrt{5}$

Since, $\sqrt{18}+\sqrt{41}\ne \sqrt{5}$ that is $AB+BC\ne AC$.

Hence, the given points are not collinear.