Question

Determine, by distance formula, whether the points ($-$6,$-$2), (2,3 and (10,8) are collinear.

Answer 1

Let L(–6, –2), M(2, 3) and N(10, 8) be three points.
From these points, x₁ = – 6, y₁ = –2, x₂ = 2, y₂ = 3, x₃ = 10, y₃ = 8.
By the distance formula, we get:

$$\begin{gathered} \mathrm{LM}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2} \\ =\sqrt{{\left[2-\left(-6\right)\right]}^2+[3-(-2){]}^2} \\ =\sqrt{(8{)}^2+(5{)}^2} \\ =\sqrt{64+25} \\ =\sqrt{89} \end{gathered}$$

$$\begin{gathered} \mathrm{MN}=\sqrt{(x_3-x_2{)}^2+(y_3-y_2{)}^2} \\ =\sqrt{(10-2{)}^2+(8-3{)}^2} \\ =\sqrt{(8{)}^2+(5{)}^2} \\ =\sqrt{64+25} \\ =\sqrt{89} \end{gathered}$$

$$\begin{gathered} \mathrm{LN}=\sqrt{(x_3-x_1{)}^2+(y_3-y_1{)}^2} \\ =\sqrt{{\left[10-(-6)\right]}^2+[8-(-2){]}^2} \\ =\sqrt{(16{)}^2+(10{)}^2} \\ =\sqrt{256+100} \\ =\sqrt{356} \\ =2\sqrt{89} \end{gathered}$$

$$\begin{gathered} \mathrm{Here},\mathrm{LM}+\mathrm{MN}=\sqrt{89}+\sqrt{89}=2\sqrt{89} \\ \mathrm{And}\mathrm{LN}=2\sqrt{89} \\ \mathrm{i}.\mathrm{e}.\mathrm{LM}+\mathrm{MN}=\mathrm{LN} \\ \therefore \mathrm{Points}\mathrm{L},\mathrm{M}\mathrm{and}\text{N}\mathrm{are}\mathrm{collinear}. \end{gathered}$$