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  3. Eight points are equally spaced on a circle. If 4 of the 8 points are to be chosen at random, what is the probability that a qua

Eight points are equally spaced on a circle. If 4 of the 8 points are to be chosen at random, what is the probability that a qua

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问题 Eight points are equally spaced on a circle. If 4 of the 8 points are to be chosen at random, what is the probability that a quadrilateral having the 4 points chosen as vertices will be a square?
选项 A、1/70
B、1/35
C、1/7
D、1/4
E、1/2
答案B
解析 For questions involving geometry, it is often helpful to draw a figure representing the information in the question as accurately as possible. The figure below shows a circle with 8 equally spaced points, labeled A through H, and quadrilateral BCDH, which is one of the many quadrilaterals that have 4 of the 8 equally spaced points as vertices.

The probability that a quadrilateral having the 4 points chosen as vertices will be a square is equal to the following fraction.
     the number of squares that can be drawn using 4 of the 8 points as vertices
       the number of quadrilaterals that can be drawn using 4 of the 8 points as vertices
To calculate the desired probability, you need to determine the number of squares and the number of quadrilaterals that can be drawn using 4 of the 8 points as vertices.
    To determine the number of quadrilaterals, first note that since the 8 points lie on a circle, every subset of 4 of the 8 points determines a unique quadrilateral. Therefore, the number of quadrilaterals that can be drawn using 4 of the 8 points as vertices is equal to the number of ways of choosing 4 points from the 8 points shown. The number of ways of choosing 4 points from the 8 points shown(also called the number of combinations of 8 objects taken 4 at a time)is equal to. You can calculate the value of this expression as follows.

= 70
Thus, there are 70 quadrilaterals that can be drawn using 4 of the 8 points as vertices.
    Because the points are equally spaced around the circle, there are only 2 squares that can be drawn using 4 of the 8 points as vertices, namely ACEG and BDFH, as shown in the following figures.

Therefore, the probability that the quadrilateral will be a square is 2/70, or 1/35, and the correct answer is Choice B.
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