If R = (0,0) and T = (2a,0) in equilateral triangle RST what are the coordinates of S in terms of a?

Question

dumbcow · Answer

distance of RT is:
$$d = \sqrt{(2a)^{2} +0^{2}} = 2a$$
let x,y be coordinates of S
then for RS
$$2a = \sqrt{x^{2}+y^{2}}$$
for TS
$$2a = \sqrt{(x-2a)^{2}+y^{2}}$$
Using substitution
$$\sqrt{x^{2}+y^{2}} = \sqrt{(x-2a)^{2}+y^{2}}$$
$$x^{2} +y^{2} = (x-2a)^{2}+y^{2}$$
$$x^{2} = (x-2a)^{2}$$
$$x^{2} = x^{2}-4ax +4a^{2}$$
$$4ax = 4a^{2}$$
$$x = a$$
Then
$$2a = \sqrt{a^{2}+y^{2}}$$
$$4a^{2} = a^{2}+y^{2}$$
$$y^{2} = 3a^{2}$$
$$y = \sqrt{3}a$$

S= (a,sqrt(3)a)