Question

An equilateral triangle has one vertex at (0, 0) and another at $(3,\sqrt{3})$. What are the coordinates of the third vertex?

• A. $(0,2\sqrt{3})$ only
• B. $(3,-\sqrt{3})$ only
• C. $(0,2\sqrt{3})$ or $(3-\sqrt{3})$
• D. Neither $(0,2\sqrt{3})$ nor $(3,-\sqrt{3})$

Answer 1

The correct option is A $(0,2\sqrt{3})$ only

The 3rd vertex need to be at $\sqrt{3^2+{\sqrt{3}}^2}=\sqrt{12}$ distance from both the vertices and should be inclined at ${60}^{\circ }$ to the line joining these 2 coordinates. Clearly, $(0,2\sqrt{3})$ is $\sqrt{12}$ distance away from both the points whereas $(3,-\sqrt{3})$ is not at said distance away from both the points. Also, line joining $(0,2\sqrt{3})$ and $(0,0)$ is inclined at ${60}^{\circ }$ with the line joining $(0,0)$ and $(3,\sqrt{3})$. Hence, out of the given options, $(0,2\sqrt{3})$ is the required point.