HMMT 二月 2020 · 几何 · 第 2 题

HMMT February 2020 — Geometry — Problem 2

Let ABC be a triangle with AB = 5, AC = 8, and ∠ BAC = 60 . Let U V W XY Z be a regular hexagon that is inscribed inside ABC such that U and V lie on side BA , W and X lie on side AC , and Z lies on side CB . What is the side length of hexagon U V W XY Z ?

解析

Let ABC be a triangle with AB = 5, AC = 8, and ∠ BAC = 60 . Let U V W XY Z be a regular hexagon that is inscribed inside ABC such that U and V lie on side BA , W and X lie on side AC , and Z lies on side CB . What is the side length of hexagon U V W XY Z ? Proposed by: Ryan Kim 40 Answer: 21 Solution: Let the side length of U V W XY Z be s . We have W Z = 2 s and W Z ‖ AB by properties of regular hexagons. Thus, triangles W CZ and ACB are similar. AW V is an equilateral triangle, so we have AW = s . Thus, using similar triangles, we have W C AC 8 − s 8 = = ⇒ = , W Z AB 2 s 5 40 so 5(8 − s ) = 8(2 s ) = ⇒ s = . 21 C Y X W Z A V U B