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## Answers

Let the third vertex of the equilaterla triangle be (x, y)

Distance between (0, 0) and (x, y) = Distance between (0, 0) and (3, √3) = Distance between (x, y) and (3, √3)

√(x2 + y2) = √(32 + 3) = √[(x - 3)2 + (y - √3)2]

x2 + y2 = 12

x2 + 9 - 6x + y2 + 3 - 2√3y = 12

24 - 6x - 2√3y = 12

- 6x - 2√3y = - 12

3x + √3y = 6

x = (6 - √3y) / 3

⇒ [(6 - √3y)/3]2 + y2 = 12

⇒ (36 + 3y2 - 12√3y) / 9 + y2 = 12

⇒ 36 + 3y2 - 12√3y + 9y2 = 108

⇒ - 12√3y + 12y2 - 72 = 0

⇒ -√3y + y2 - 6 = 0

⇒ (y - 2√3)(y + √3) = 0

⇒ y = 2√3 or - √3

If y = 2√3, x = (6 - 6) / 3 = 0

If y = -√3, x = (6 + 3) / 3 = 3

So, the third vertex of the equilateral triangle = (0, 2√3) or (3, -√3).