If a vertex, the circumcenter, and the centroid of a triangle are (0, 0), (3,4), and (6, 8), respectively, then the triangle must be (a)   a right-angled triangle (b)   an equilateral triangle (c)   an isosceles triangle (d)   a right-angled isosceles triangle

Explanation

We have vertex A(0,0) circumcentre O(3,4) and centroid G(6,8). Point O lies on the perpendicular bisector (OD) of side BC and point G lies on the median AD.
Now, A O and G are collinear. So median AD and perpendicular bisector (OD) of BC coincide. Thus, AD is the perpendicular bisector of BC. Therefore, triangle ABC is isosceles.