HMMT 十一月 2012 · 冲刺赛 · 第 21 题

HMMT November 2012 — Guts Round — Problem 21

[ 11 ] ABC is a triangle with AB = 7, BC = 10, and CA = 13. Point D lies on segment BC such that DC = 2 BD . Point E lies on segment AD such that AE = 4 ED . Point F lies on segment AC such that F C = 5 AF . Find the area of △ EF C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND

解析

[ 11 ] √ 80 3 Answer: By shared bases, we know that 9 ( ) ( ) ( ) ( ) ( ) ( ) 5 5 4 5 4 2 [ EF C ] = [ AEC ] = [ ADC ] = [ ABC ] . 6 6 5 6 5 3 √ √ √ 80 3 By Heron’s formula, we find that [ ABC ] = (15)(8)(2)(5) = 20 3, so [ AEC ] = 9 Guts Round