If two vertices of an equilateral triangle be (0, 0) and (3, ), find the co - ordinates of the third vertex.

Given that A(0, 0) and B(3, ) are two vertices of an equilateral triangle.

Let us assume C(x, y) be the third vertex of the triangle.

We have AB = BC = CA

We know that the distance between the two points (x₁, y₁) and (x₂, y₂) is .

Now,

⇒ BC = CA

⇒ BC² = CA²

⇒ (3 - x)² + ( - y)² = (x - 0)² + (y - 0)²

⇒ x² - 6x + 9 + 3 + y² - 2y = x² + y²

⇒ 6x = 12 - 2y

..... - (1)

⇒ AB = BC

⇒ AB² = BC²

⇒ (0 - 3)² + (0 - )² = (3 - x)² + ( - y)²

⇒ 9 + 3 = 9 - 6x + x² + 3 - 2y + y²

From (1)

⇒

⇒ 48y² - 48√3y - 288 = 0

⇒ y²–√3y –6 = 0

⇒ y² - 2√3y + √3y - 6 = 0

⇒ y(y - 2√3) + √3(y - 2√3) = 0

⇒ (y + √3)(y - 2√3) = 0

⇒ y + √3 = 0 (or) y - 2√3 = 0

⇒ y = - √3 (or) y = 2√3

From (1), for y =

⇒ x = 3

From (1), for y = 2

⇒ x = 0

∴ The third vertex of equilateral triangle is (0, 2√3) and (3, √3).