In DS questions, we cannot trust the figure - only the data stated in the question stem and statements counts.
The triangle looks like an equilateral triangle, but this fact is not mentioned in the question stem. If the statement(s) somehow fix the triangle as equilateral, then we can find the area - using the 30:60:90 1:√3:2 ratio, we can find the side of the triangle from the height, and from that find the area. The issue is therefore whether the statement(s) are enough to prove that ABC is equilateral.
With this in mind, the answer is indeed C: each of the statements deals with one of the "halves" of big triangle ABC, but you need both in order to show that both halves are 30:60:90 triangles.
Stat. (1): If ABD= 60 degrees, then the left triangle is a 30:60:90 triangle. But as nightreader's later post shows, this does not tell us that the right half is also a 40:60:90 triangle, since we do not know that the height AD is also a bisector.
Stat (2): from AC=12 and the height, we can find DC using the pythagorean theorem. Guess what, it'll come down to 6. The right triangle ADC satisfies the 6:6√3:6*2 ratio of a 30:60:90 triangle, and we can therefore infer that its angles are indeed 30, 60 and 90, respectively. This takes care of the right triangle, but alone is insufficient, as it says nothing of the left side.
Combined: each of the triangle "halves" is a 30:60:90 triangle, and both share the same height AD=6√3. Therefore, the two trinagles are equal, the triangle ABC is indeed an equilateral triangle, and the area can be found using the equilateral formula side^2*√3/4 = 12√3/4 = 3√3. Sufficient.
it can be anything
