HMMT 二月 2004 · 冲刺赛 · 第 27 题

HMMT February 2004 — Guts Round — Problem 27

[9] A regular hexagon has one side along the diameter of a semicircle, and the two opposite vertices on the semicircle. Find the area of the hexagon if the diameter of the semicircle is 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND

解析

A regular hexagon has one side along the diameter of a semicircle, and the two opposite vertices on the semicircle. Find the area of the hexagon if the diameter of the semicircle is 1. √ Solution: 3 3 / 26 The midpoint of the side of the hexagon on the diameter is the center of the circle. Draw the segment from this center to a vertex of the hexagon on the circle. This segment, whose length is 1 / 2, is the hypotenuse of a right triangle whose legs have √ 2 lengths a/ 2 and a 3, where a is a side of the hexagon. So 1 / 4 = a (1 / 4 + 3), so 2 a = 1 / 13. The hexagon consists of 6 equilateral triangles of side length a , so the area √ √ 2 of the hexagon is 3 a 3 / 2 = 3 3 / 26. a 7