Question

Use the distance formula to prove that the point $A(-2,3),B(1,2)$ and $C(7,0)$ are collinear.

Answer 1

Using distance formula,

$D=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

AB$=\sqrt{{(1+2)}^2+{(2-3)}^2}=\sqrt{9+1}=\sqrt{10}$

BC$=\sqrt{{(7-1)}^2+{(0-2)}^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10}$

CA$=\sqrt{{(7+2)}^2+{(0-3)}^2}=\sqrt{81+9}=\sqrt{90}=3\sqrt{10}$

Thus, $AC=AB+BC$ from above

Hence, $A,B,C$ are collinear.