[img]2022m10x/ct_ve01204002j_0052_221018[/img]In the figure above, what is the

游客2025-02-27 32

问题

In the figure above, what is the perimeter of △PQF ?
(1) The length of segment PT is 2.
(2) The length of segment RS is

.

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 Determine the perimeter of APQR by determining PQ + QR + PR.
(1) It is given that PT= 2. Since APTQ is a 45°-45°-90° triangle, it follows that QT = 2 and PQ =

. Since △QTR is a 30°—60°—90° triangle and QT = 2, it follows that QR = 4 and TR =

. PT+ TR = PR, so PR, QR, and PQ are known and the perimeter of △PQR can be determined; SUFFICIENT.
(2) It is given that RS =

, but no information is given to determine TS. If, for example, TS = V3, then AQTR is a 30°—60°—90° triangle with TS+SR=TR =

. It follows that QT = 2 and QR = 4. Also, △PTQ is a 45°-45°-90° triangle with QT= 2. It follows that PT= 2 (hence PR = 2 +

) and PQ =

, so the perimeter of the triangle is

. However, if TS =

, then △QTR is a 30°—60°—90° triangle with TS+SR=TR =

. It follows that QT= 3, and QR = 6. Also, APTQ is a 45°—45°—90° triangle with QT = 3. It follows that PT= 3 (hence PR = 3 +

) and PQ =

, so the perimeter of the triangle is

; NOT sufficient.
The correct answer is A;
statement 1 alone is sufficient.

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