A vertex of an equilateral triangle is `2,3` and the opposite side is `x+y=2.` Find the equations of other sides.

Options

A. `(2 pm sqrt(3))x -y = 1 pm2sqrt(3)`
B. `(2pm sqrt(3))x -y = 1pm 2 sqrt(3)`
C. `(1pm 2sqrt(3)) x-y = 2 pm sqrt(3)`
D. `(2pm sqrt(3))x +y=1 pm 2sqrt(3)`

Answer

A

Explanation

Let A(2,3) be one vertex and x +y =2 be the opposited side of an equilateral triangle. Clearly, remainig two sides pass through the point A(2,3) and make an angle of `60^(@)` with x+ y =2

Let m be the slope of x + y= 2 . Then m =- 1. So the equations of the othere two sides are
`y-3=(-1-tan 60^(@))/(1-tan 60^(@))(x-2)and y -3=(-1+tan 60^(@))/(1+tan 60^(@))(x-2)`
`rArr y -3 =(-(1+sqrt(3)))/(1-sqrt(3))(x-2) and y-3=(sqrt(3)-1)/(sqrt(3)+1) (x-2)`
`rArr (2+sqrt(3))x -y = 1+ 2sqrt(3) and (2-sqrt(3))x-y = 1-2sqrt(3)`