2. 外语
  3. DYou are given that the circle has a diameter of 10, and from the figure you can assume that points A, B, C, and D lie on the ci

DYou are given that the circle has a diameter of 10, and from the figure you can assume that points A, B, C, and D lie on the ci

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答案D
解析 You are given that the circle has a diameter of 10, and from the figure you can assume that points A, B, C, and D lie on the circle in the order shown. However, because figures are not necessarily drawn to scale, you cannot assume anything else about the positions of points A, B, C, and D on the circle. Therefore, to get an idea of how various possible positions of these four points could affect the area of quadrilateral ABCD, it is a good idea to see how the figure can vary but still have points A, B, C, and D in the same order as in the figure above.
    One way that you might vary the figure is to evenly space the four points along the circle, as shown below.

    Another way is to draw points A and C opposite each other, with points B and D close to point C, as shown below.

    From these figures you can draw some basic conclusions about the area of ABCD.
    If points A and C are opposite each other, with points B and D very close to point C, the area of quadrilateral ABCD is very close to 0. Clearly, the area can be less than 40(Quantity B).
    If points A, B, C, and D are evenly spaced, the area is not close to 0. How does the area compare with 40? To calculate the area of ABCD, draw the diameters AC and BD in the figure. The two diameters are perpendicular bisectors of each other, so they divide ABCD into four right triangles, as shown.

    The area of each of the right triangles is(1/2)(5)(5), or 12.5. Thus, the area of
ABCD is(4)(12.5), or 50.
    Since the area of the quadrilateral in the first figure is less than 40 and the area of the quadrilateral in the second figure is greater than 40, the relationship cannot be determined from the information given. The correct answer is Choice D.
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