The length of the height AD of the triangle ABC is half the length of the side BC. Can angle A be obtuse?

Suppose that our triangle is inscribed in a circle, then the height of the AO will be equal to the radius of the circle and equal to half the diameter of the aircraft. In this case, the angle BAC is based on the diameter and is then equal to 90.

When mixing the AO height along the BC diameter, the BA1C angle will be less than the BAC angle and, therefore, the ABC angle has a maximum value of 90, and cannot be obtuse.

Answer: Angle A cannot be obtuse.