So from the question, you should be able to tell that the diagram looks like this:
OR like this:
We are only told that there are 4 points of the circle--so all of them could be in any order. At first, you might think that the 4 points must be in order alphabetically as you go clockwise around the circle. But you cannot make this assumption since the question does not name the circle ABCD. Rather, it says there are 4 points on the circle.As such, there are two possible situations you need to consider:
1) When the two segments overlap
2) When the two segments do not overlap
Technically, you can even argue the case where CD is inside the bigger segment AB. But then it would be obvious which segment is longer than the other one. So for the purposes of this GMAT question, you should just consider the overlap and non-overlap situations.We are told X< pi and Y < pi. Why are they telling us that? Is that pointless additional information or does it actually mean something?
Well, they tell us the radius is 1. We know that the circumference is always some number times pi. It's pi * diameter. So that means the circumference is 2pi. Well, if you imagine an upper half of the circle and lower half of the circle, then each half has length of pi.
We're told that x < pi and y< pi. That means the distance between A and B must be within the upper hemisphere--not exceeding the dotted line in the middle. Likewise, the distance between C and D must be within the lower hemisphere--not exceeding the dotted line in the middle. Now, let's look at statement #1: Angle ADB is acute.
This basically tells us that there is NO overlap between the two segments. Why? Look at the overlap image. Clearly, angle ADB is obtuse (>90 degrees). So this overlap diagram is no good if we use statement #1.
Now look at the non-overlap image. Angle ADB is acute.So really, statement #1 is telling us that there is no overlap.
But we still don't know which segment is longer than the other one. We only know there is no overlap.Now let's look at statement #2:
Angle ADB is > angle CAD.
In the Overlap diagram
, it's tough to see how this will help us, if at all. We don't know which segment is longer.
In the non-overlap diagram
, let's pretend ADB and CAD are the same. Now try to shift point B down-right so ADB > CAD while keeping everything the same. What do you notice? Well, you notice the distance between A and B is getting bigger while the distance between C and D stays the same.
So that's good, but that's only when the segments do not overlap. When they do overlap, we still don't know which segment is longer.
What if you combined (1) and (2)?
Statement #1 basically tells us that we are only dealing with the non-overlap diagram.
Statement #2 tells us that for the non-overlap diagrams, one segment is always greater than the other segment.
This is exactly what we're looking for! The question asks "is x>y"? Well, by knowing statement #2 that ADB > CAD---combined with statement #1 that there is NO overlap between the two segments, we know that yes, the length of X > length of Y. So statement #1 combined with statement #2 is enough to answer the question with only 1 solution. When #1 is no good by itself and #2 is no good by itself, but when combined we can narrow down everything to one correct answer, then the final answer is (C).
Overall, I think this question is more complicated than a typical GMAT question you would see. Even if you got a question this wordy, it would definitely come with a diagram to explain exactly what is going on. I see complications with this question if you swap points C and D--because all the naming of angles become unclear.
In any event, take from this explanation what you can in terms of identifying possible situations and piecing together information.
Hope that helps!