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HMMT 十一月 2012 · 冲刺赛 · 第 13 题

HMMT November 2012 — Guts Round — Problem 13

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 9 ] Triangle ABC has a perimeter of 18. ∠ A = 60 , and BC = 7. Find the area of the triangle.
解析

英文解析

  1. [ 9 ]
    √
    Answer: 6 3 Let AB be x . Then, AC = 18 − 7 − x = 11 − x . Using the law of cosines gives:
    2 2 ° 2
    x + (11 − x ) − 2 x (11 − x ) cos 60 = 7 .
    3 x − 33 x + 72 = 0 .2
    x = 3 or 8 .
    Therefore, AB = 8 and AC = 3 or AB = 3 and AC = 8. In both cases, the area of the triangle is:
    √
    °1
    · 8 · 3 sin 60 = 6 3 . 2