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

Explanation

Given line is
`x+y-2 = 0 " "(1)`
Its slope is `m_(1) = -1`
Let the slope of the line be m which makes an angle of ` 60^(@)` with line in (1). Then,
`tan 60^(@) = |(m_(1)-m)/(1+m_(1)m)| " or " sqrt(3) = |(-1-m)/(1-m)|`
`" or " sqrt(3) = |(1+m)/(m-1)| " or " (1+m)/(m-1) = +-sqrt(3)`
`"or "1+m= +-sqrt(3)(m-1)`
` "or " m=(sqrt(3)+1)/(sqrt(3)-1), (sqrt(3)-1)/(sqrt(3)+1)`
`= 2+ sqrt(3), 2-sqrt(3)`
Therefore, the equations of other two sides of the triangle are
`y-3 = (2+sqrt(3)) (x-2)`
` "and "y-3 = (2-sqrt(3)) (x-2)`