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  3. In the figure above, the shaded region represents the front of an upright wooden frame around the entrance to an amusement park

In the figure above, the shaded region represents the front of an upright wooden frame around the entrance to an amusement park

admin2022-10-18  89
问题
In the figure above, the shaded region represents the front of an upright wooden frame around the entrance to an amusement park ride. If RS = meters, what is the area of the front of the frame?
(1) x = 9 meters
(2) ST = meters
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D
解析 The front of the frame is in the shape of an equilateral triangle with sides of length x with an opening in the shape of an equilateral triangle with sides of length y, where y < x. The area of the front of the frame is the area of the larger triangle minus the area of the smaller triangle.
Note that the height of an equilateral triangle with sides of length s is given by and the area is given by both of which are easily derived using the ratios for the sides of a 30°-60°-90° triangle.
(1)    Given that x = 9, it follows that RT， the height of the large equilateral triangle, is and RT=RS+ST, it follows that ST= where ST is the height of the small equilateral triangle. Therefore, y = 4. With the side lengths of both triangles known, the area of the front of the frame can be determined; SUFFICIENT.
(2)    Given that ST= it follows that y = 4.
Also, since RT=RS+ ST and RS = it follows that RT= and x = 9. With the side lengths of both triangles known, the area of the front of the frame can be determined; SUFFICIENT.
The correct answer is D;
each statement alone is sufficient.
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