AMC8 2020

AMC8 2020 · Q25

AMC8 2020 · Q25. It mainly tests Systems of equations, Coordinate geometry.

Rectangles $R_1$ and $R_2$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

下面的矩形 $R_1$ 和 $R_2$ 以及正方形 $S_1,\,S_2,\,$ 和 $S_3$ 组合成一个宽 $3322$ 单位、高 $2020$ 单位的矩形。$S_2$ 的边长是多少单位？

(A) 651 651

(B) 655 655

(D) 662 662

(E) 666 666

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let the side length of each square $S_k$ be $s_k$. Then, from the diagram, we can line up the top horizontal lengths of $S_1$, $S_2$, and $S_3$ to cover the top side of the large rectangle, so $s_{1}+s_{2}+s_{3}=3322$. Similarly, the short side of $R_2$ will be $s_1-s_2$, and lining this up with the left side of $S_3$ to cover the vertical side of the large rectangle gives $s_{1}-s_{2}+s_{3}=2020$. We subtract the second equation from the first to obtain $2s_{2}=1302$, and thus $s_{2}=\boxed{\textbf{(A) }651}$.

设每个正方形 $S_k$ 的边长为 $s_k$。从图中，正方形 $S_1$、$S_2$、$S_3$ 的顶边总长覆盖大矩形的顶边，故 $s_{1}+s_{2}+s_{3}=3322$。类似地，$R_2$ 的短边为 $s_1-s_2$，与 $S_3$ 左边对齐覆盖大矩形竖边，故 $s_{1}-s_{2}+s_{3}=2020$。两式相减得 $2s_{2}=1302$，从而 $s_{2}=\boxed{\textbf{(A) }651}$。

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