HMMT 二月 2006 · 冲刺赛 · 第 6 题
HMMT February 2006 — Guts Round — Problem 6
题目详情
英文原题
- [6] Suppose ABC is a triangle such that AB = 13 , BC = 15, and CA = 14. Say D is themidpoint of BC , E is the midpoint of AD , F is the midpoint of BE , and G is the midpointof DF . Compute the area of triangle EF G .
IX HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUNDth
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x x 2
解析
英文解析
- Suppose ABC is a triangle such that AB = 13 , BC = 15, and CA = 14. Say D isthe midpoint of BC , E is the midpoint of AD , F is the midpoint of BE , and G is themidpoint of DF . Compute the area of triangle EF G .
Answer:21
√4
Solution: By Heron’s formula, [ ABC ] = 21(21 − 15)(21 − 14)(21 − 13) = 84.
[ DEF ] [ BDE ] [ ABD ]
Now, unwinding the midpoint conditions yields [ EF G ] = = = =
2 4 8
[ ABC ]
84 21 = = .
16 16 4