HMMT 二月 2010 · 冲刺赛 · 第 13 题

HMMT February 2010 — Guts Round — Problem 13

[ 8 ] A triangle in the xy -plane is such that when projected onto the x -axis, y -axis, and the line y = x , the results are line segments whose endpoints are (1 , 0) and (5 , 0), (0 , 8) and (0 , 13), and (5 , 5) and (7 . 5 , 7 . 5), respectively. What is the triangle’s area?

解析

[ 8 ] A triangle in the xy -plane is such that when projected onto the x -axis, y -axis, and the line y = x , the results are line segments whose endpoints are (1 , 0) and (5 , 0), (0 , 8) and (0 , 13), and (5 , 5) and (7 . 5 , 7 . 5), respectively. What is the triangle’s area? 17 Answer: Sketch the lines x = 1, x = 5, y = 8, y = 13, y = 10 − x , and y = 15 − x . The 2 triangle has to be contained in the hexagonal region contained in all these lines. If all the projections are correct, every other vertex of the hexagon must be a vertex of the triangle, which gives us two possibilities for the triangle. One of these triangles has vertices at (2 , 8), (1 , 13), and (5 , 10), and has 17 an area of . It is easy to check that the other triangle has the same area, so the answer is unique. 2