AMC10 2013 B

AMC10 2013 B · Q12

AMC10 2013 B · Q12. It mainly tests Probability (basic), Polygons.

Let S be the set of sides and diagonals of a regular pentagon. A pair of elements of S are selected at random without replacement. What is the probability that the two chosen segments have the same length?

设 $S$ 是一个正五边形的边和对角线的集合。从 $S$ 中不放回地随机选取一对元素。所选两条线段长度相等的概率是多少？

(A) $\frac{2}{5}$ $\frac{2}{5}$

(B) $\frac{4}{9}$ $\frac{4}{9}$

(D) $\frac{5}{9}$ $\frac{5}{9}$

(E) $\frac{4}{5}$ $\frac{4}{5}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

The five sides of the pentagon are congruent, and the five congruent diagonals are longer than the sides. Once one segment is selected, 4 of the 9 remaining segments have the same length as that segment. Therefore the requested probability is 4/9.

五边形的五条边全等，五条全等对角线比边长。选定一条线段后，剩下 9 条线段中有 4 条与它长度相同。因此所求概率是 $\frac{4}{9}$。

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