Question

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

Answer 1

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)