AMC10 2014 A

AMC10 2014 A · Q23

AMC10 2014 A · Q23. It mainly tests Area & perimeter, Transformations.

A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B : A$?

一张长方形纸片的长度是宽度的 $\sqrt{3}$ 倍，面积为 $A$。纸片沿相对长度方向分成三个相等部分，然后如图从第一个分隔线到对侧第二个分隔线画一条虚线。然后沿这条虚线平折成一个新形状，面积为 $B$。$B : A$ 的比值为多少？

(A) 1 : 2 1 : 2

(B) 3 : 5 3 : 5

(D) 3 : 4 3 : 4

(E) 4 : 5 4 : 5

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Without loss of generality, assume that the rectangle has dimensions $3$ by $\sqrt{3}$. Then the fold has length $2$, and the overlapping areas are equilateral triangles each with area $\frac{\sqrt{3}}{4}\cdot 2^2$. The new shape has area $3\sqrt{3}-\frac{\sqrt{3}}{4}\cdot 2^2=2\sqrt{3}$, and the desired ratio is $2\sqrt{3}:3\sqrt{3}=2:3$.

答案（C）：不失一般性，设矩形的尺寸为 $3$ 乘 $\sqrt{3}$。则折痕长度为 $2$，重叠部分是两个全等的正三角形，每个面积为 $\frac{\sqrt{3}}{4}\cdot 2^2$。新图形的面积为 $3\sqrt{3}-\frac{\sqrt{3}}{4}\cdot 2^2=2\sqrt{3}$，所求比为 $2\sqrt{3}:3\sqrt{3}=2:3$。

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