2. 外语
  3. In rectangular region PQRS above, T is a point on side PS. If PS = 4, what is the area of region PQRS? (1) △QTR is equilateral.

In rectangular region PQRS above, T is a point on side PS. If PS = 4, what is the area of region PQRS? (1) △QTR is equilateral.

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问题
In rectangular region PQRS above, T is a point on side PS. If PS = 4, what is the area of region PQRS?
(1) △QTR is equilateral.
(2) Segments PT and TS have equal length.
选项 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.
答案A
解析 It is given that PQRS is a rectangle and PS = 4. The area of PQRS can be determined if and only if PQ can be determined.
(1)
It is given that △QTR is equilateral. If is the height of △QTR as shown above, then △QUT is a 30-60-90° triangle with UR = 2. Using the ratios for 30-60-90 triangles, TU= . Since PQ = TU, the area of PQRS can be determined; SUFFICIENT.
(2)    Given that PT= TS, PQ could be any positive number. Thus, it is not possible to determine PQ and therefore not possible to determine the area of PQRS; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
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