Question

Show that the following points form an equilateral triangle. $(\sqrt{3},2),(0,1)$ and $(0,3)$

Answer 1

Let, the given points are $A(\sqrt{3},2),B(0,1)$ and $C(0,3)$

Now

$AB=\sqrt{(\sqrt{3}-0{)}^2+(2-1{)}^2}$

$=\sqrt{(\sqrt{3}{)}^2+(1{)}^2}$

$=\sqrt{3+1}=2$

$BC=\sqrt{(0-0{)}^2+(1-3{)}^2}$

$=\sqrt{2^2}$

$=2$

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

$=\sqrt{(\sqrt{3}{)}^2+(1{)}^2}$

$=\sqrt{3+1}=2$

$\therefore$ In $△ABC$ , $AB=BC=CA$

Hence it is an equilateral triangle