Question

The coordinates of three snooker balls are 𝐀(−𝟐, 𝟔), 𝐁(𝟎, 𝟐), and 𝐂(𝟐, −𝟐). Determine if they are collinear or not.

Answer 1

Find AB, BC and AC using the distance formula.

Check AB+BC=AC for collinearity

Let A$(x_1,y_1)=(-2,6)$
B$(x_2,y_2)=(0,2)$ and
C$(x_3,y_3)=(2,-2)$.
Apply the distance formula for AB, BC, and AC.
AB = $\sqrt{(}(0+2{)}^2+(2-6{)}^2)$
AB = $\sqrt{2}0$
BC = $\sqrt{(}(2-0{)}^2+(-2-2{)}^2)$
BC = $\sqrt{2}0$
AC = $\sqrt{(}(2+2)2+(-2-6)2)$
AC = √80 = 2√20.
Verify whether AB + BC = AC satisfies or not
AB + BC = $\sqrt{2}0$ + $\sqrt{2}0$ = 2$\sqrt{2}0$
Hence, AB + BC = AC