Question

Examine whether points (3, 3) (-4, -1) and (3,-5) are the vertices of an isosceles triangle.

Answer 1

Let L(3, 3), M(–4, –1) and N(3, –5) be the three points.
From these points, x₁ = 3, y₁ = 3, x₂ = – 4, y₂ = –1, x₃ = 3, y₃ = –5.
By the distance formula,
$$\begin{gathered} \mathrm{LM}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2} \\ =\sqrt{(-4-3{)}^2+(-1-3){}^2} \\ =\sqrt{(-7{)}^2+(-4{)}^2} \\ =\sqrt{49+16} \\ =\sqrt{65} \end{gathered}$$

$$\begin{gathered} \mathrm{MN}=\sqrt{(x_3-x_2{)}^2+(y_3-y_2{)}^2} \\ =\sqrt{[3-(-4){]}^2+[-5-(-1){]}^2} \\ =\sqrt{(7{)}^2+(-4{)}^2} \\ =\sqrt{49+16} \\ =\sqrt{65} \end{gathered}$$

$$\begin{gathered} \mathrm{LN}=\sqrt{(x_3-x_1{)}^2+(y_3-y_1{)}^2} \\ =\sqrt{(3-3{)}^2+(-5-3{)}^2} \\ =\sqrt{(0{)}^2+(-8{)}^2} \\ =\sqrt{0+64} \\ =\sqrt{64} \\ =8 \end{gathered}$$

$$\begin{gathered} \mathrm{Here},\mathrm{LM}=\mathrm{MN}=\sqrt{65}. \\ \mathrm{As}\text{the}\mathrm{two}\mathrm{sides}\mathrm{of}△\mathrm{LMN}\mathrm{are}\mathrm{equal},\text{it}\mathrm{is}\mathrm{an}\mathrm{isosceles}\mathrm{triangle}. \end{gathered}$$