Manhattan GMAT Challenge Problem of the Week – 10 September 2012
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In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD? (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 AD is 2.
(2) The angle between side AB and side BC is 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.
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.
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