Question

在一个等边三角形的一边上作一个正方形。在正方形的一条不相邻的边上作一个正五边形，如图所示。在五边形的一条不相邻的边上作一个正六边形。用同样方式继续作正多边形，直到作出正八边形。所得图形的外边界有多少条边？

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?

$21$

$23$

$25$

$27$

$29$

答案：B

知识点：正多边形基本计数

难度评级：1120

解答：

三角形和八边形在链的两端，所以各有一条边成为共享边而不在外边界上。正方形、五边形、六边形和七边形在中间，所以各有两条边成为共享边。

所得图形有条边。 $\begin{aligned} &(3-1)+(4-2) \\ &\quad {}+(5-2)+(6-2) \\ &\quad {}+(7-2)+(8-1)=23 \end{aligned}$

所以正确答案是 B。

The triangle and octagon are at the ends of the chain, so each loses one side to a shared edge. The square, pentagon, hexagon, and heptagon are in the middle, so each loses two sides to shared edges.

The resulting polygon has $\begin{aligned} &(3-1)+(4-2) \\ &\quad {}+(5-2)+(6-2) \\ &\quad {}+(7-2)+(8-1)=23 \end{aligned}$ sides.

Thus, B is the correct answer.

2009 AMC 8 第 9 题

答案：B

难度评级：1120

解答：

其他年份的第 9 题