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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 & 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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