AMC12 2023 B

AMC12 2023 B · Q9

AMC12 2023 B · Q9. It mainly tests Absolute value, Coordinate geometry.

What is the area of the region in the coordinate plane defined by $| | x | - 1 | + | | y | - 1 | \le 1$?

坐标平面中由 $| | x | - 1 | + | | y | - 1 | \le 1$定义的区域的面积是多少？

(A) 2 2

(B) 8 8

(D) 15 15

(E) 12 12

Answer

Correct choice: (B)

正确答案：（B）

Solution

First consider, $|x-1|+|y-1| \le 1.$ We can see that it is a square with a side length of $\sqrt{2}$ (diagonal $2$). The area of the square is $\sqrt{2}^2 = 2.$ Next, we insert an absolute value sign into the equation and get $|x-1|+||y|-1| \le 1.$ This will double the square reflecting over x-axis. So now we have $2$ squares. Finally, we add one more absolute value and obtain $||x|-1|+||y|-1| \le 1.$ This will double the squares as we reflect the $2$ squares we already have over the y-axis. Concluding, we have $4$ congruent squares. Thus, the total area is $4\cdot2 =$ $\boxed{\text{(B) 8}}$

首先考虑$|x-1|+|y-1| \le 1$。我们可以看到这是一个对角线为$2$的正方形，边长为$\sqrt{2}$。正方形的面积为$\sqrt{2}^2 = 2$。接下来，在方程中插入一个绝对值符号，得到$|x-1|+||y|-1| \le 1$。这会将正方形关于$x$轴反射，加倍。现在我们有两个正方形。最后，再加一个绝对值，得到$||x|-1|+||y|-1| \le 1$。这会将已有的两个正方形关于$y$轴反射，加倍。结论，我们有4个全等的正方形。因此，总面积为$4\cdot2 = \boxed{\text{(B) 8}}$

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