Find coordinates of point C in a equilateral triangle

Question

How to find the coordinates of point C in a equilateral triangle, where $A=(-2,2)$ and $B=(6,2)$.

https://i.stack.imgur.com/TXjjG.png

Thanks in advance

Answer 1

Hint:

1) note that $\overline {AB}=8$.

2) Note that $C$ isa point on the line orthogonal to $AB$ in the middle point, so it is a point on the vertical line $x=2$. This means that $C$ has coordinates $C=(2,y)$

3) You want $\overline{AB}=\overline{AC}$ and this gives an equation that you can solve for $y$ and gives two results (why?)

---

You have $\overline{AC}=\sqrt{(2+2)^2+(y-2)^2}=\sqrt{16+(y-2)^2}$, so the equation that you have to solve is: $$ 64=16+y^2+4-4y \quad \iff \quad y^2-4y-44=0 $$

can you solve it?