Question

ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.

Answer 1

ABC is an equilateral triangle.

∴ AC = BC  and AB = BC

⇒ AC² = BC²   and AB² = BC²

(x - 1)² + (y - 1)² = (x + 1)² + (y + 1)²

⇒ x² + 1 - 2x + y² + 1 - 2y = x² + 1 + 2x + y² + 1+ 2y

⇒ - 4 x - 4y = 0

⇒ - 4 x = 4y

⇒ x = - y             ...........(1)

(1 + 1)² + (1 + 1)² = (x + 1)² + (y + 1)²

⇒ 8 = x² + 1 + 2x + y² + 1 + 2y

⇒ 8 = y² + 1 + 2x + y² + 1 + 2y

⇒ 2y² - 6 = 0

⇒ y² = 3

⇒ y = ${\displaystyle \pm \sqrt{3}}$

From (1)

∴ x = ${\displaystyle \pm \sqrt{3}}$