Let the circle with centre at origin pass through the vertices of an equilateral triangle A B C . If A (2,4) , then the length of the median through A is

Step-by-step solution

Given the circle centered at O(0,0) passes through the vertices of equilateral triangle ABC, with vertex A at (2,4) . For an equilateral triangle, the centroid G coincides with the circumcenter, hence G = (0,0) . The centroid divides the median AM in the ratio 2:1 , so AG = 2 3 AM . The distance between A(2,4) and G(0,0) is AG = (2-0)^2 + (4-0)^2 = 4 + 16 = 20 = 2 5 . Substituting into the centroid relation, 2 5 = 2 3 AM , so AM = 3 2 2 5 = 3 5 . Thus, the length of the median through A is 3 5 units, corresponding