2023 AMC 12B 第 9 题

Question

坐标平面中，由下列不等式定义的区域面积是多少？

||x|-1|+||y|-1|\le 1\text{？}

What is the area of the region in the coordinate plane defined by

||x|-1|+||y|-1|\le 1?

2

8

4

15

12

该区域关于两个坐标轴都对称，所以用

|x|

和

|y|

来考虑

The region is symmetric across both axes, so work with

|x|

and

|y|

在第一象限中，

|x-1|+|y-1|\le 1

是面积为

2

的菱形；共有

4

份

In the first quadrant

|x-1|+|y-1|\le 1

is a diamond of area

2;

there are

4

copies

将

x,y

替换为

|x|,|y|

，条件

|x-1|+|y-1|\le 1

描述的是以

(1,1)

为中心、对角线长为

2

的菱形，因此面积为

2

。它完全位于第一象限（只在单个点接触坐标轴），所以关于两条坐标轴反射后得到

4

个互不重叠的副本。总面积为

4\times 2=8

。

因此，正确答案是 B。

Replacing

x,y

by

|x|,|y|,

the condition

|x-1|+|y-1|\le 1

describes a diamond centered at

(1,1)

with diagonals of length

2,

hence area

2.

It lies entirely in the first quadrant (touching the axes only at single points), so reflecting across the two axes produces

4

disjoint copies. The total area is

4\times 2=8.

Thus, the correct answer is B.

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