HMMT 二月 2017 · 几何 · 第 4 题
HMMT February 2017 — Geometry — Problem 4
题目详情
- Let ABCD be a convex quadrilateral with AB = 5 , BC = 6 , CD = 7, and DA = 8. Let M, P, N, Q be 2 2 the midpoints of sides AB, BC, CD, DA respectively. Compute M N − P Q .
解析
- Let ABCD be a convex quadrilateral with AB = 5 , BC = 6 , CD = 7, and DA = 8. Let M, P, N, Q be 2 2 the midpoints of sides AB, BC, CD, DA respectively. Compute M N − P Q . Proposed by: Sam Korsky Answer: 13 2 Draw in the diagonals of the quad and use the median formula three times to get M N in terms of the 2 diagonals. Do the same for P Q and subtract, the diagonal length terms disappear and the answer is 2 2 2 2 BC + DA − AB − CD = 13 2