Explanation

We have quite an abundance of points and line segments here. If we were short on time in the Quantitative section and saw this question, we would glance at this, punch in a (D), and move on. One thing to keep in mind while practicing is that it's a perfectly fine habit to work on mastering every question, considering variations, meditating on confusing points, and so on, but guessing at proper times tends to be the practice of people who score 700+ and even 760+ on the GMAT. Anyway, in this question, we want the area of the circle and the area of the square, so that we can find the difference. The radius of the circle and the side of a length of the square will get us to that point. The circle is centered on D, so BD or DF would give us the radius and hence the area of the circle, for one thing. On to the statements, separately first.

Statement (1) allows us to conclude that , and since point C bisects AF, we know that and the area of the square is 400. That leaves the question of whether we can determine the radius of the circle. Indeed, if and , then the diameter of the circle is . We are able to determine the area of the circle and will be able to answer the question definitively. Sufficient.

Statement (2) is somewhat like the information of Statement (1) backwards. EF is half a radius, so it allows us to determine the radius of the circle. And that means we have BF, which along with the measure of AB given, lets us know the measure of a side of the square. So Statement (2) is also sufficient.

The correct answer is (D).