题目信息
When opened and lying flat, a birthday card is in the shape of a regular hexagon. The card must be folded in half along 1 of its diagonals before being placed in an envelope for mailing. Assuming that the thickness of the folded card will not be an issue, will the birthday card fit inside a rectangular envelope that is 4 inches by 9 inches?
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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正确答案：
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Geometry Polygons
As shown in the figure above, a regular hexagon with sides of length s can be partitioned into six equilateral triangles. Using these triangles, it is possible to determine the length of each diagonal (2s), the height, shown as a dashed line, of each triangle
, the area of each triangular region 
, and the area of the hexagonal region 
.
When the birthday card is folded in half along one of the diagonals it has the shape shown below.
Given s = 4, the maximum width of the birthday card is 2s = 8, which is less than the width of the envelope, and its height is
= 
, which is less than the height of the envelope because 
< 
= 4. Thus, the birthday card will fit in the envelope; SUFFICIENT.
Given that the surface area of the card when folded is less than 36 square inches, it follows that 
< 36, which simplifies to 
. If s = 4, then the birthday card will fit in the envelope, as shown in (1) above. However, if s = 5, then s < 
(note that 625 = 54 < (
)4 = 768), but the maximum width of the birthday card will be 2s = 10, and the card will not fit in the envelope; NOT sufficient.
The correct answer is A;statement 1 alone is sufficient.
As shown in the figure above, a regular hexagon with sides of length s can be partitioned into six equilateral triangles. Using these triangles, it is possible to determine the length of each diagonal (2s), the height, shown as a dashed line, of each triangle
, the area of each triangular region , and the area of the hexagonal region .When the birthday card is folded in half along one of the diagonals it has the shape shown below.
Given s = 4, the maximum width of the birthday card is 2s = 8, which is less than the width of the envelope, and its height is
= , which is less than the height of the envelope because < = 4. Thus, the birthday card will fit in the envelope; SUFFICIENT.
Given that the surface area of the card when folded is less than 36 square inches, it follows that < 36, which simplifies to . If s = 4, then the birthday card will fit in the envelope, as shown in (1) above. However, if s = 5, then s < (note that 625 = 54 < ()4 = 768), but the maximum width of the birthday card will be 2s = 10, and the card will not fit in the envelope; NOT sufficient.The correct answer is A;statement 1 alone is sufficient.
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