HMMT 二月 2018 · 几何 · 第 2 题
HMMT February 2018 — Geometry — Problem 2
题目详情
- Points A, B, C, D are chosen in the plane such that segments AB , BC , CD , DA have lengths 2, 7, 5, 12, respectively. Let m be the minimum possible value of the length of segment AC and let M be the maximum possible value of the length of segment AC . What is the ordered pair ( m, M )?
解析
- Points A, B, C, D are chosen in the plane such that segments AB , BC , CD , DA have lengths 2, 7, 5, 12, respectively. Let m be the minimum possible value of the length of segment AC and let M be the maximum possible value of the length of segment AC . What is the ordered pair ( m, M )? Proposed by: Kevin Sun Answer: (7 , 9) By the triangle inequality on triangle ACD , AC + CD ≥ AD , or AC ≥ 7. The minimum of 7 can be achieved when A , C , D lie on a line in that order. By the triangle inequality on triangle ABC , AB + BC ≥ AC , or AC ≤ 9. The maximum of 9 can be achieved when A , B , C lie on a line in that order. This gives the answer (7 , 9).