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Question
ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.
Solution
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 = `+- sqrt 3`
From (1)
∴ x = `+- sqrt 3`
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