In rectangular region PQRS above, T is a point on side PS. If PS = 4,

VERITAS PREP OFFICIAL SOLUTION:

Question Type: What Is the Value? This question asks for the area of the rectangle PQRS.

Given information in the question stem or diagram: The diagram is given and importantly one side is defined with PS = 4, so both PS and QR = 4. To find the area you need to somehow determine either RS or PQ from the information in the statements. The question is really: What is RS or PQ?

Statement 1: QTR is an equilateral triangle. If this is true, then with QR as the base, the height of the equilateral triangle is equivalent to the side that you are trying to determine (PQ or RS). Since the base of QTR is 4 you can determine the necessary height with your knowledge of 30–60–90 triangles or simply with your knowledge that if you know one thing about an equilateral triangle, then you know everything! While you do not need to calculate it, the height would be 2√3 as it is the long leg in a 30–60–90 triangle formed by 1/2 the base of QTR (2), the hypotenuse QT (4) and the height ( 2√3 ) which is equivalent to PQ or RS. Statement 1 is sufficient, so the answer is A or D.

Statement 2: In this difficult statement, the testmakers are playing with a common trick. They have polluted your brain with the first statement and want you to assume that if “segments PT and TS have equal lengths,” then QTR must again be equilateral. However, this statement does nothing to help you determine the length of the sides (QP and RS) because it only proves that QTR is isosceles. There is no limit put on the lengths of PQ and RS (because you do not know the angle of TQR and TRQ) with this statement, so it is not sufficient. Remember: One of the keys to success in Data Sufficiency is to consciously avoid assumptions, but that can be hard when you are set up so nicely to make assumptions with the other statement. Statement 2 is not sufficient, so the correct answer is A.