Question

In the given figure, ABC is an equilateral triangle. Find the co-ordinates of A.

Answer 1

Given:

ABC is an equilateral triangle with vertices B(2, 0) and C(6, 0) lying on the x-axis

Formula:

\[\text{Side length of triangle} = x_C - x_B \text{ and height of an equilateral triangle} = \frac{\sqrt{3}}{2} \times \text{side}\]

Solution:

Side = 6 − 2 = 4  [Calculating the length of side BC using coordinates of B and C]

\[\text{x-coordinate of A} = \frac{2 + 6}{2} = 4 \quad [\text{Using the midpoint formula since the altitude bisects the base}]\]

\[\text{Height of triangle} = \frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3} \quad [\text{Calculating the height (y-coordinate of A) of the equilateral triangle}]\]

\[\therefore \text{y-coordinate of A} = 2\sqrt{3} \quad [\text{Equating the height to the vertical distance from the x-axis}]\]

\[\therefore {(4, 2\sqrt{3})}\]