# Manhattan GMAT Challenge Problem of the Week – 10 September 2012

Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people that enter our challenge, the better the prizes!

## Question

In quadrilateral

ABCD, sidesABandBCeach have length √2, while sideCDhas length 2. What is the area of quadrilateralABCD? (For this problem, “quadrilateral” means any closed figure with four straight sides in a plane, with each side touching exactly two other sides at their endpoints.)(1) The length of side

ADis 2.

(2) The angle between sideABand sideBCis 90°.A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

## Answer

You are told that “quadrilateral” *ABCD* has these side lengths: √2 (= *AB*), √2 (= *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 a completely equivalent possibility (e.g., four line segments intersecting in a plane at exactly four points) 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.**

**Special Announcement:** If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner receives a number of our our Strategy Guides. The more people enter, the better the prize. Provided the winner gives consent, we will post his or her name on our Facebook page.

## 2 comments

Lalit on September 10th, 2012 at 4:24 am

Pls help. Still did not get (looks like missing link between Q49 and Q51 )

If I join AC, Area(ABCD) = Area (ACD) - Area(ABC)

and we can very well find out Area(ACD and Area(ABC)

ashok on September 20th, 2012 at 10:26 am

Case1 in the above explanation gives u an area of 4 SqU while in case2, it is 1 SqU... So, no definite answer and hence E