The line AD, perpendicular to the median BM of the triangle ABC, divides the BAC angle in half. Find AC side if side AB is 3

AD for triangle ABM is both median and height. And this is the property of the median for an isosceles triangle. Therefore, the triangle ABM is isosceles with the base BM.
By the definition of an isosceles triangle, AB = AM.
Because BM is the median for triangle ABC, therefore AM = MC (as defined by the median).
Then AC = AM * 2. As we found out earlier, AM = AB => AC = AB * 2 = 3 * 2 = 6.
Answer: AC = 6.