There was a lot of confusion and anxiety regarding last week’s Challenge Problem Showdown. And for good reason; it was a difficult problem! In fact, only 15% of submitted answers were correct, making this the most difficult Challenge Problem Showdown in several years (by the way, you can purchase our complete Challenge Problem Showdown Archive here).
With this in mind, here is the solution to the Challenge Problem.
You are told that quadrilateral ABCD has these side lengths: âˆš2 (= AB), âˆš2 (= BC), and 2 (= CD). You don’t know any angles, so imagine that you have hooked together three poles representing the three sides that you know. The poles are connected by flexible hinges at B and C:
You don’t know how long the fourth side (AD) is, so imagine that the fourth side is a rubber band that can stretch and shrink as you adjust the angles. ABCD forms a quadrilateral, defined in this problem as any closed figure with four straight sides in a plane, with each side touching exactly two other sides. In other words, the sides can’t cross over each other, but you can have an indentation, such as the one shown below at point B:
So you know, the term quadrilateral is actually defined on the GMAT more narrowly than it is in typical geometry classes (and in this problem) to include only convex quadrilaterals, in which every internal angle is less than 180 degrees. So when the GMAT uses the term quadrilateral, you can ignore concave possibilities such as the one shown above.
Statement (1): NOT SUFFICIENT. The fourth side (AD) is fixed in length, but you can collapse or open up the kite to make different areas. In many cases, knowing the 4 sides of a quadrilateral does not determine the area of the quadrilateral (even restricting yourself to convex quadrilaterals), since you can often change the angles and therefore the area without changing the sides. Consider that a square and a typical rhombus can both have the same side lengths, but the rhombus has less area because you have collapsed the square somewhat.
Statement (2): NOT SUFFICIENT. Fixing the angle between two of the sides still leaves you freedom to swing the third side (and therefore the fourth side) in various directions, leading to different areas.
Statements (1) and (2) TOGETHER: STILL NOT SUFFICIENT! Here are the two possible pictures of the quadrilateral, pictures that satisfy both constraints but that have different areas:
To be fair, if you used the GMAT’s normal definition of quadrilateral, the concave possibility on the right would be outlawed, and C would be the right answer. So if you picked C, don’t feel bad!
That said, does that mean you should never think about concave shapes such as the one encountered here? Well, the GMAT is very clever, and the writers might avoid using the term quadrilateral while making you think about an equivalent possibility (e.g., four line segments intersecting in a plane at exactly four distinct points, end to end, enclosing a planar figure) that has absolutely no such restriction against concavity. Always pay close attention to the restrictions as given; if terms are defined or redefined in the problem, those definitions will be critical.
The correct answer is E.