In an isosceles triangle ABC, the medians intersect at point O. Find the distance from point

In an isosceles triangle ABC, the medians intersect at point O. Find the distance from point O to apex A of this triangle, if AB = AC = 25 cm, BC = 14 cm.

In an isosceles triangle: the median drawn to the base is also the bisector and height. Consider triangle ABM, M is the point of intersection of the median from vertex A to the side BC and the side BC. Angle AMB = 90 degrees, since AB = AC, AM – median => height. BM = MC = 1 / 2BC = 7. By the Pythagorean theorem: AB ^ 2 = AM ^ 2 + BM ^ 2 => AM = root of (25-7) (25 + 7) = root of 576 = 24. By property medians: the intersection of the medians of a triangle divides each median by a ratio of 2/1 from the apex of the triangle. Hence, AO = 2/3 AM = 2/3 * 24 = 16.
Answer: AO = 16.