Question

The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

Answer 1

Let ABC be the given equilateral triangle with side 2a.

Accordingly, AB = BC = CA = 2a

Assume that base BC lies along the y-axis such that the mid-point of BC is at the origin.

i.e., BO = OC = a, where O is the origin.

Now, it is clear that the coordinates of point C are (0, a), while the coordinates of point B are (0, –a).

It is known that the line joining a vertex of an equilateral triangle with the mid-point of its opposite side is perpendicular.

Hence, vertex A lies on the y-axis.

On applying Pythagoras theorem to ΔAOC, we obtain

(AC)² = (OA)² + (OC)²

⇒ (2a)² = (OA)² + a ²

⇒ 4a ² – a ² = (OA)²

⇒ (OA)² = 3a ²

⇒ OA =

∴Coordinates of point A =

Thus, the vertices of the given equilateral triangle are (0, a), (0, –a), and  or (0, a), (0, –a), and.