Question:

Show that $\displaystyle A(-2,1)$ , $\displaystyle B(5,-2)$ and $\displaystyle C(3,3)$ are vertices of an equilateral triangle.

Attempt:

Distance between points $\displaystyle A(-2,1)$ and $\displaystyle B(5,-2)$:

$\displaystyle AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$\displaystyle AB = \sqrt{(5-(-2))^2 + (-2-1)^2}$

$\displaystyle AB = \sqrt{49 + 9}$

$\displaystyle AB = \sqrt{58}$

Distance between points $\displaystyle B(5,-2)$ and $\displaystyle C(3,3)$:

$\displaystyle BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$\displaystyle BC = \sqrt{(3-5)^2+(3-(-2))^2}$

$\displaystyle BC = \sqrt{4 + 25}$

$\displaystyle BC = \sqrt{29}$

Distance between points $\displaystyle A(-2,1)$ and $\displaystyle C(3,3)$:

$\displaystyle AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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

$\displaystyle AC = \sqrt{25 + 4}$

$\displaystyle AC = \sqrt{29}$

I have found out the distance between the points. How can I find out if it is a right angled triangle?