Here's a diagram of the figure described.
Notice what I did, which is to use a trick that works well on many GMAT triangle questions.
Depending on the situation, you may be able to use any triangle that fits the description. In many situations everything has to work as long as the triangle fits all of the constraints. In this situation, I made the angle at A a right angle, because doing that does not change any key parameters of the question and it makes everything come out more easily.
Statement 1 tells us the area of triangle ABX. Notice that in the version of the figure that I drew, the height of ABX is half that of ABC, while they share the same base.
This means that the area of ABC is twice that of ABX. So the area of ABC is 2 x 32 = 64.
Now we are getting somewhere.
If the length of segment CR is 1/2 of that of segment CX which is 1/2 of that of segment CA, then CR is 1/4 CA.
So the height of triangle RCS is 1/4 of that of ABC.
The other thing you need to know is that triangles which share an angle and have two proportional sides that meet at that angle are similar triangles. This concept is rarely incorporated into GMAT questions.
In any case what this means is that since RC is 1/4 of AC and SC is 1/4 of BC, triangle RCS is similar to triangle ABC, with all dimensions of RCS being 1/4 of those of ABC.
Since we know the area of ABC is 64 and we know the height of RCS is 1/4 of that of ABC and the base of RCS is also 1/4 of that of ABC, we can calculate the area of RCS.
So Statement 1 is sufficient.
Statement 2 gives us one of the heights of triangle ABC. We can't calculate area using just the height.
So Statement 2 is insufficient.
The correct answer is
A.
