In triangle ABC, the median AD is drawn from vertex A to side BC, and it is given that the length AD equals one half of side BC. What is the measure of angle BAC (in degrees)?

Introduction / Context:
This geometry question explores a special property of medians in a triangle. It connects the length of a median to the type of triangle, specifically to right angled triangles. Recognizing this property allows us to find the measure of an angle without heavy calculation. Such questions often appear in geometry sections of school exams and competitive tests.

Given Data / Assumptions:

Triangle ABC is given.
AD is the median from vertex A to side BC, so D is the midpoint of BC.
The length of the median AD satisfies AD = (1/2) * BC.
We need to find angle BAC in degrees.

Concept / Approach:
A well known theorem states that in a right angled triangle, the median drawn from the right angle to the hypotenuse is equal to half the length of the hypotenuse. The converse is also true: if the median from a vertex to the opposite side has length equal to half the opposite side, then the triangle is right angled at that vertex, and the opposite side is the hypotenuse. Here BC plays the role of the hypotenuse, and AD is the median from A, so we expect angle BAC to be a right angle.

Step-by-Step Solution:
AD is a median, so D is the midpoint of BC and BD = DC = BC/2.Given AD = BC/2.Thus AD = BD = DC, which means triangle BDC is isosceles with BD = DC and point A lies such that AD is also equal to these segments.Consider triangle ABC. If angle BAC is a right angle, then BC is the hypotenuse.In a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices.So if ABC is right angled at A, the midpoint D of BC will satisfy AD = BD = CD = BC/2.This matches the given condition AD = BC/2.By the converse of this result, the fact that the median from A to BC is half of BC implies that triangle ABC is right angled at A.Therefore angle BAC = 90°.

Verification / Alternative check:
We can also verify using coordinates and distance formulas. Place B and C so that BC is horizontal and let D be the origin. If BD = DC = r, then B and C are at (−r, 0) and (r, 0). Let A be at (0, h). Then AD = h, BD = DC = r, and BC = 2r. Condition AD = BC/2 implies h = r. The distances AB and AC are both √(r^2 + r^2) = r√2, so triangle ABC is isosceles with equal legs AB and AC and vertex A at (0, r). The angle at A is 90° because the vectors AB and AC are perpendicular. This confirms the theoretical argument.

Why Other Options Are Wrong:
Angles 30°, 45°, 60°, and 75° correspond to non right angled triangles where the median from A to BC is not exactly half of BC. They would not satisfy the geometric property that the midpoint of the hypotenuse is equidistant from all three vertices. Therefore none of these angles fit the given length condition for AD.

Common Pitfalls:
Students sometimes try to apply the medians length formula or Apollonius theorem directly without remembering the simpler special property of right triangles. Others confuse the roles of sides and medians or assume that the equality AD = BC/2 holds in many types of triangles, which is not true. Recognizing this as a right triangle configuration simplifies the problem enormously.

Final Answer:
The measure of angle BAC in triangle ABC is 90°.