1965 AMC 12 第 36 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
36.
给定两条不同的直线 和 。从 上一点向 作垂线，再从该垂线的垂足向 作垂线。从第二条垂线的垂足再向 作垂线，如此无限继续。第一、第二条垂线段的长度分别为 和 ；当垂线条数无限增加时，各垂线段长度之和趋于一个极限。此极限为：
Given distinct straight lines and From a point in a perpendicular is drawn to from the foot of this perpendicular a line is drawn perpendicular to From the foot of this second perpendicular a line is drawn perpendicular to and so on indefinitely. The lengths of the first and second perpendiculars are and respectively. Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is:
小提示：
两条固定直线依次形成的直角三角形相似
Successive right triangles formed by the two fixed lines are similar
大提示：
垂线段长度构成首项为 、公比为 的等比数列
The perpendicular lengths form a geometric sequence with first term and ratio
解答：
每个新的直角三角形都有相同的锐角，所以垂线段长度构成等比数列。由于前两项为 ， 公比为 。 收敛意味着 ， 因而总和为
因此，正确答案是 E。
Each new right triangle has the same acute angle, so the perpendicular lengths form a geometric sequence. Since the first two lengths are the common ratio is Convergence implies and the sum is
Therefore, the correct answer is E.