One vertex of the equilateral triangle with centroid at origin and one side $$x + y − 2 = 0$$​ is:

Correct option is B

Given:
Centroid of the equilateral triangle is at (0, 0).
One side of the triangle is x + y − 2 = 0.
Formula used:
The centroid divides the median in the ratio $$2 : 1$$​ from the vertex.
​For an equilateral triangle, the median is perpendicular to the side and
passes through the centroid.
Solution:
The line x + y − 2 = 0 has slope −1.
So, the median will have slope 1.
Equation of median through origin:
y = x
The median meets the given side at its midpoint.
Find the midpoint of the side x + y − 2 = 0 lying on y = x.
Substitute y = x:
x + x − 2 = 0
2x = 2
x = 1, y = 1
So midpoint of the side is (1, 1).
The centroid divides the median in the ratio $$2 : 1.$$​​
Hence, the vertex lies on the median such that:
​$$OG : GM = 2 : 1$$​​
So the vertex is at: $$(2, 2)$$​​
The correct answer is (b) (2, 2).