AMC 8 2009 Problem 9
AMC8 2009 年第 9 题
四边形与多边形★★☆☆☆
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
在一个等边三角形的一边上作一个正方形，然后在此正方形的另一条非相邻边上作一个正五边形，如下图所示。再在此五边形的一条非相邻边上作一个六边形。按照这种方法不断作出正多边形，直到作出一个八边形。那么最终形成的多边形有多少条边？
- A.21
- B.23
- C.25
- D.27
- E.29
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
Of the six shapes used to create the polygon, the triangle and octagon are adjacent to the others on one side, and the others are adjacent on two sides. In the triangle and octagon 3+8-2(1)=9 sides are on the outside of the final polygon. In the other shapes 4+5+6+7-4(2) = 14 sides are on the outside. The resulting polygon has 9+14 = 23 sides.
We can quickly see a pattern if we draw out the other shapes. Every shape will have two of its sides taken out except the triangle and octagon. We can then make the expression 2+2+3+4+5+7 which is 23
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