HMMT 二月 2000 · 几何 · 第 4 题
HMMT February 2000 — Geometry — Problem 4
题目详情
- Find the total area of the non-triangle regions in the gure b elo w (the shaded area). a/4 b/3 b a/3 a
解析
- Noti e that in general, when there is a re tangle of side length x and y , the area of the non-triangle regions ( reated b y dra wing a line onne ting the midp oin t of t w o opp osite lines and a line onne ting t w o opp osite orners - see diagram for examples) 3 is simply of the original area of the b o x, sin e the area of the ex luded triangles are 4 1 y x 1 y x xy xy ab 3
- = 2 = , so the desired area is simply ab = ab , as desired. 2 2 2 2 2 2 2 4 4 4 So, if w e onsider the four re tangles making up the large re tangle that are divided a b this w a y (the one with dimensions x , et .), w e an sa y that the total shaded area is 4 3 3 3 of the total area of the re tangle - that is, ab . 4 4 1 1 3 9 Æ Æ