HMMT 十一月 2018 · 冲刺赛 · 第 26 题

HMMT November 2018 — Guts Round — Problem 26

[ 13 ] Points E, F, G, H are chosen on segments AB, BC, CD, DA , respectively, of square ABCD . Given that segment EG has length 7, segment F H has length 8, and that EG and F H intersect inside ABCD ◦ at an acute angle of 30 , then compute the area of square ABCD .

解析

[ 13 ] Points E, F, G, H are chosen on segments AB, BC, CD, DA , respectively, of square ABCD . Given that segment EG has length 7, segment F H has length 8, and that EG and F H intersect inside ABCD at an acute angle of 30 , then compute the area of square ABCD . Proposed by: Kevin Sun 784 Answer: 19 0 0 0 0 0 0 Rotate EG by 90 about the center of the square to E G with E 2 AD and G 2 BC . Now E G 0 0 and F H intersect at an angle of 60 . Then consider the translation which takes E to H and G to I . Triangle F HI has F H = 8 , HI = 7 and \ F HI = 60 . Furthermore, the height of this triangle is the side length of the square. Using the Law of Cosines, p p 2 2 F I = 7 7 · 8 + 8 = 57 . By computing the area of F HI in two ways, if h is the height then p p 1 1 3 ⇥ 57 ⇥ h = ⇥ ⇥ 7 ⇥ 8 . 2 2 2 28 2 784 p Then h = and the area of the square is h = . 19 19