AMC8 2009

AMC8 2009 · Q9

AMC8 2009 · Q9. It mainly tests Basic counting (rules of product/sum), Polygons.

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 regular 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条边上构造一个正方形。在正方形的一条不相邻边上构造一个正五边形，如图所示。在正五边形的一条不相邻边上构造一个正六边形。以同样的方式继续构造正多边形，直到构造出八边形。最终得到的的多边形有多少条边？

(A) 21 21

(B) 23 23

(D) 27 27

(E) 29 29

Answer

Correct choice: (B)

正确答案：（B）

Solution

One side of the triangle and one side of the octagon will each touch one other polygon. Two sides of the other polygons will touch other polygons. Make a table and add the appropriate number of sides. The resulting polygon has 2 + 2 + 3 + 4 + 5 + 7 = 23 sides.

三角形的一条边和八边形的一条边各与一个其他多边形相触。其余多边形的两条边与其它多边形相触。制作表格并添加适当的边数。最终多边形有 $2 + 2 + 3 + 4 + 5 + 7 = 23$ 条边。

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