AMC12 2022 A

AMC12 2022 A · Q3

AMC12 2022 A · Q3. It mainly tests Area & perimeter.

Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?

五个矩形$A$、$B$、$C$、$D$和$E$排列成一个正方形，如下图所示。这些矩形的尺寸分别是$1\times6$、$2\times4$、$5\times6$、$2\times7$和$2\times3$。（图未按比例绘制。）中间阴影的矩形是哪一个？

(A) A A

(B) B B

(D) D D

(E) E E

Answer

Correct choice: (B)

正确答案：（B）

Solution

The area of this square is equal to $6 + 8 + 30 + 14 + 6 = 64$, and thus its side lengths are $8$. The sum of the dimensions of the rectangles are $2 + 7 + 5 + 6 + 2 + 3 + 1 + 6 + 2 + 4 = 38$. Thus, because the perimeter of the square is $32$, the rectangle on the inside must have a perimeter of $6 \cdot 2 = 12$. The only rectangle that works is $\boxed{\textbf{(B) }B}$.

这个正方形的面积等于$6 + 8 + 30 + 14 + 6 = 64$，因此其边长为$8$。矩形尺寸之和为$2 + 7 + 5 + 6 + 2 + 3 + 1 + 6 + 2 + 4 = 38$。因此，正方形的周长为$32$，内部矩形的周长必须为$6 \cdot 2 = 12$。唯一符合的矩形是$\boxed{\textbf{(B) }B}$。

Topics

GE.007 Area & perimeter