Question

Show that the points (− 3, − 3), (3, 3) and $\left(-3\sqrt{3},3\sqrt{3}\right)$ are the vertices of an equilateral triangle.

Answer 1

Let the given points be A(− 3, − 3), B(3, 3) and C$\left(-3\sqrt{3},3\sqrt{3}\right)$. Now
$$\begin{gathered} AB=\sqrt{{\left(-3-3\right)}^2+{\left(-3-3\right)}^2}=\sqrt{{\left(-6\right)}^2+{\left(-6\right)}^2} \\ =\sqrt{36+36}=\sqrt{72}=6\sqrt{2} \\ BC=\sqrt{{\left(3+3\sqrt{3}\right)}^2+{\left(3-3\sqrt{3}\right)}^2} \\ =\sqrt{9+27+18\sqrt{3}+9+27-18\sqrt{3}}=\sqrt{72}=6\sqrt{2} \end{gathered}$$
$$\begin{gathered} AC=\sqrt{{\left(-3+3\sqrt{3}\right)}^2+{\left(-3-3\sqrt{3}\right)}^2}=\sqrt{{\left(3-3\sqrt{3}\right)}^2+{\left(3+3\sqrt{3}\right)}^2} \\ =\sqrt{9+27-18\sqrt{3}+9+27+18\sqrt{3}} \\ =\sqrt{72}=6\sqrt{2} \end{gathered}$$
Hence, the given points are the vertices of an equilateral triangle.