1972 AMC 12 第 30 题

Question

一张宽

6

英寸的长方形纸按图折叠，使一个角落在对边上。用角

\theta

表示的折痕

L

长度（英寸）为：

A rectangular piece of paper

6

inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease

L

in terms of angle

\theta

is:

3\sec^2\theta\csc\theta

6\sin\theta\sec\theta

3\sec\theta\csc\theta

6\sec\theta\csc^2\theta

以上皆非

None of these

设长方形纸的高为

h

，并利用折叠产生的相等长度

Let

h

be the height of the rectangular sheet and use the equal lengths created by the fold

由图可得

\frac{6}{h}=\sin(2\theta)

和

\frac{L}{h}=\sec\theta

The diagram gives

\frac{6}{h}=\sin(2\theta)

and

\frac{L}{h}=\sec\theta

设纸张的高为

h

。把折起的角关于折痕反射所产生的全等直角三角形给出

\frac6h=\sin(2\theta)=2\sin\theta\cos\theta\text{，}

所以

h=\frac{3}{\sin\theta\cos\theta}

。它们还给出

\frac{L}{h}=\sec\theta

。因此

\begin{aligned} L&=h\sec\theta\\ &=\frac{3\sec\theta} {\sin\theta\cos\theta}\\ &=3\sec^2\theta\csc\theta \end{aligned}\text{。}

因此，正确答案是 A。

Let

h

be the sheet’s height. The congruent right triangles created by reflecting the folded corner across the crease give

\frac6h=\sin(2\theta)=2\sin\theta\cos\theta,

so

h=\frac{3}{\sin\theta\cos\theta}.

They also give

\frac{L}{h}=\sec\theta.

Hence

\begin{aligned} L&=h\sec\theta\\ &=\frac{3\sec\theta} {\sin\theta\cos\theta}\\ &=3\sec^2\theta\csc\theta. \end{aligned}

Therefore, the correct answer is A.

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