2006 AMC 12A 第 6 题

Question

如图，

8 \times 18

的矩形

ABCD

被切成两个全等的六边形，使这两个六边形可以无重叠地重新拼成一个正方形。

y

是多少？

The

8 \times 18

rectangle

ABCD

is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is

y?

6

7

8

9

10

两个六边形重新拼成面积为

8\cdot 18

的正方形。

The two hexagons reassemble into a square of area

8\cdot 18

阶梯形切线把宽分成三个相等的水平部分，它们的和为

18

。

The staircase splits the width into three equal horizontal pieces that sum to

18

两个六边形组成一个面积为

8 \cdot 18 = 144

的正方形，所以正方形边长为

12

。

阶梯形切线把宽分成三个长度为

y

的相等水平部分，它们合起来跨过整个宽度：

y + y + y = 18

，所以

y = 6

。（两个竖直台阶各上升

12 - 8 = 4

，形成正方形多出的高度。）

因此，正确答案是 A。

The two hexagons form a square of area

8 \cdot 18 = 144,

so the square has side

12.

The staircase cut splits the width into three equal horizontal pieces of length

y,

which together span the full width:

y + y + y = 18,

so

y = 6.

(The two vertical steps each rise

12 - 8 = 4,

building the extra height of the square.)

Thus, the correct answer is A.

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