In triangle ABC, the median AD is drawn from vertex A to side BC. If AD is 6 cm and the length of side BC is 12 cm, then what is the measure of angle CAB?

Introduction / Context:
This geometry problem focuses on a special property of right triangles involving the median drawn to the hypotenuse. It checks whether you remember a key fact: in a right triangle, the median drawn from the right angle vertex to the hypotenuse has a very specific relationship with the hypotenuse. Recognizing this property allows you to conclude which angle is a right angle without lengthy calculations.

Given Data / Assumptions:

• Triangle ABC is given.
• AD is a median drawn from vertex A to side BC, so D is the midpoint of BC.
• Length of median AD = 6 cm.
• Length of side BC = 12 cm.
• We are asked to find the measure of angle CAB.
• The triangle is assumed to lie in a standard Euclidean plane with usual triangle properties.

Concept / Approach:
In any triangle, a median joins a vertex to the midpoint of the opposite side. In a right triangle there is an important special result: the median drawn from the right angle to the hypotenuse is exactly half the hypotenuse. That means if a triangle has a hypotenuse of length h and a median from the right angle vertex to the hypotenuse of length h / 2, then that vertex must be the right angle. We will compare the given median length and half of BC to decide whether angle CAB is the right angle.

Step-by-Step Solution:
Step 1: D is the midpoint of BC since AD is a median. Step 2: The length of BC is given as 12 cm. Step 3: Compute half of BC: BC / 2 = 12 / 2 = 6 cm. Step 4: The given length of the median AD is also 6 cm. Step 5: Therefore AD = BC / 2. Step 6: In a triangle, if the median from a vertex to the opposite side equals half of that opposite side, then that opposite side is the hypotenuse and the vertex is a right angle. Step 7: Here, BC acts as the hypotenuse and A is the right angle vertex. Step 8: So angle CAB is the angle at A, and it must be 90°.

Verification / Alternative check:
Another way to think about this is to recall that in a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices. Thus, if AD is the segment from the right angle vertex A to the midpoint D of BC, then AD equals the radius of the circumcircle, which is half of the hypotenuse. Since AD and BC / 2 are equal here, triangle ABC must be right angled at A. There is no need to find side lengths explicitly or to use trigonometry for this reasoning.

Why Other Options Are Wrong:
An angle of 30° or 60° is typical in special right triangles, but the question is asking which angle is the right angle, based on the median property. The values 30° and 60° do not correspond to the right angle. Similarly, 120° is an obtuse angle and cannot be part of a triangle that respects the condition about the median being half the opposite side. Only 90° is consistent with the known geometric property used in this problem.

Common Pitfalls:
Many students misinterpret the median property and think it always equals half the opposite side, which is not true. This happens only for the median drawn from the right angle to the hypotenuse in a right triangle. Others attempt to apply the Pythagoras theorem without enough side information, which leads nowhere. Some learners also confuse medians with altitudes or angle bisectors. Carefully remembering the special property about the median to the hypotenuse avoids these issues.

Final Answer:
Thus, angle CAB is a right angle and its measure is 90°.