Question

If (0, 0) and (3√3) are two vertices of an equilateral triangle, then find third vertex.

Answer 1

Let the coordinate of third vertex C of equilateral triangle is (x, y)

According to question, two vertices of equilateral triangle is A(0, 0) and B(3√3)

Squaring both sides

(2√3y)² = (12 – 6x)²

12y² = 144 + 36x²– 144x

⇒ 12(12 – x²) = 144 + 36x² – 144x [x² + y² = 12]

⇒ 144 – 12x² = 144 + 36x² – 144x

⇒ – 12x² = 36x² – 144

⇒ 36x² + 12x² – 144x = 0

⇒ 48x² – 144x = 0

⇒ 48x(x – 3) = 0

⇒ x = 0 or x – 3 = 0

⇒ x = 0 or x = 3

Hence, put x = 0 in equation (v)

x² + y² = 12

0 + y² = 12

y² = 12

y = ± 2√3

Hence, x = 0, y = ± 2√3

Put x = 3 in equation (v)

x² + y² = 12

(3)² + y² = 12

y² = 12 – 9

y = ± √3

From x = 3, y = ± √3

Hence, coordinate of third vertex is (0, 2√3), (0, -2√3), (3, √3) and (3, -√3), (3, √3) is given. 

Hence coordinate of third vertex is (0,+ 2√3) or (0, – 2.√3) or (3, -√3)