2. 外语
  3. In the figure above, if the shaded region is rectangular, what is the length of XY? (1) The perimeter of the shaded region is 24

In the figure above, if the shaded region is rectangular, what is the length of XY? (1) The perimeter of the shaded region is 24

admin2022-10-18  84
问题
In the figure above, if the shaded region is rectangular, what is the length of XY?
(1) The perimeter of the shaded region is 24.
(2) The measure of ∠XYZ is 45°.
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C
解析 Determine XY.
(1)     Letting L and W be the length and width of the shaded rectangular region, then from (1), L + W= 12. This does not give enough information about △XZY to determine XY; NOT sufficient.
(2)     From (2), △XZY is a 45°-45°-90° triangle, so XZ = YZ and XY = XZ, but no information is given to determine XZ;  NOT sufficient.
Taking (1) and (2) together and using the figure below, from (2), △XRP is a 45°-45°-90° triangle, so XR = PR=QZ= W. Likewise APQY is a 45°-45°-90° triangle, so PQ = YQ = RZ = L. It follows that XZ = XR + RZ=W+L and W+ L= 12 from (1). Likewise, YZ=YQ+QZ = L+W= 12. Since the length of the legs of △XZY are known, XY can be determined.

The correct answer is C;
both statements together are sufficient.
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