Find the third vertex of equilateral triangle, Mathematics

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If two vertices of an equilateral triangle are (0, 0) and (3, 0), find the third vertex. [Ans: 3/2 , 3/√ 3/2 or 3/2, -3√ 3/2]

Ans: OA = OB = AB

OA² = OB² = AB²

OA² = (3-0)² + 0 = 9

OB² = x² + y²

AB² = (x-3)² + y² = x² + y² - 6x + 9

OA² = OB² = AB²

OA² = OB² & OB² = AB²

9 = x² + y² ⇒ y² = 9 - x²

x² + y² - 6x + 9 = 9

x² + 9-x² - 6x + 9 = 9

6x = 9 x= 3/2

y² = 9 - (3/2)² = 9-9/4 = 36-9/4=27/4

y = ± 3√3/2

∴ Third vertex is (3/2,3√3/2) or (3/2, -3√3/2)

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