Question

If $(-4,3)$ and $(4,3)$ are two vertices of an equilateral triangle, find the coordinates of the third vertex.

Answer 1

by symmetry, it can be seen that third coordinate will be an y-axis

$|BC|=8=|AC|$

$\sqrt{(4{)}^2+(y-3{)}^2}=8$

$16+(y-3{)}^2=8$

$16+(y-3{)}^2=64$

$(y-3{)}^2=48$

$y-3=\pm 4\sqrt{3}$

$y=3\pm 4\sqrt{3}$

Therefore point is $(0,3\pm 4\sqrt{3})$