题目信息
In the figure above, if the shaded region is rectangular, what is the length of XY ?
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.
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已选答案：
正确答案：
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Geometry Triangles
Determine XY.
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 ΔXZY to determine XY; NOT sufficient. From (2), ΔXZY is a 45°−45°−90° triangle, so XZ = YZ and XY =
, but no information is given to determine XZ; NOT sufficient.
Taking (1) and (2) together and using the figure below, from (2), ΔXRP is a 45°−45°−90° triangle, so XR = PR = QZ = W. Likewise ΔPQY 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 ΔXZY are known, XY can be determined.
The correct answer is C;both statements together are sufficient.
Determine XY.
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 ΔXZY to determine XY; NOT sufficient. From (2), ΔXZY is a 45°−45°−90° triangle, so XZ = YZ and XY =
, but no information is given to determine XZ; NOT sufficient.Taking (1) and (2) together and using the figure below, from (2), ΔXRP is a 45°−45°−90° triangle, so XR = PR = QZ = W. Likewise ΔPQY 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 ΔXZY are known, XY can be determined.
The correct answer is C;both statements together are sufficient.
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