HMMT 二月 2000 · 几何 · 第 4 题
HMMT February 2000 — Geometry — Problem 4
题目详情
英文原题
- Find the total area of the non-triangle regions in the gure b elo w (the shaded area).
a/4
b/3
a/3 ba
解析
英文解析
- Noti e that in general, when there is a re tangle of side length x and y , the area ofthe non-triangle regions ( reated b y dra wing a line onne ting the midp oin t of t w oopp osite lines and a line onne ting t w o opp osite orners - see diagram for examples)
is simply of the original area of the b o x, sin e the area of the ex luded triangles are 3
1 y x 1 y x xy xy ab 34 + = 2 = , so the desired area is simply ab = ab , as desired.
2 2 2 2 2 2 2 4 4 4
So, if we onsider the four re tangles making up the large re tangle that are divideda bthis w a y (the one with dimensions x , et .), we an sa y that the total shaded area is
4 3
of the total area of the re tangle - that is, ab .33
4 4
1 1 3 9
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