Question

Show that the following points form an equilateral triangle.
$(a,0),(-a,0)$ and $(0,a\sqrt{3}$)

Answer 1

Given $A(a,0),B(-a,0),C(0,a\sqrt{3})$

$AB=\sqrt{(a-(-a){)}^2+(0-0{)}^2}=\sqrt{4a^2}=2a$

$BC=\sqrt{(0-(-a){)}^2+(a\sqrt{3}{)}^2}=\sqrt{4a^2}=2a$

$CA=\sqrt{(0-(-a){)}^2+(a\sqrt{3}{)}^2}=\sqrt{4a^2}=2a$

$\therefore AB=BC=CA$

Hence it is an equilateral triangle.