Question

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:

$3\sec^2\theta\csc\theta$

$6\sin\theta\sec\theta$

$3\sec\theta\csc\theta$

$6\sec\theta\csc^2\theta$

None of these

Answer: A

Concepts:paper folding trigonometry right triangle

Difficulty rating: 2270

Small Hint:

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

Big Hint:

The diagram gives $\frac{6}{h}=\sin(2\theta)$ and $\frac{L}{h}=\sec\theta$

Solution:

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.

1972 AMC 12 Problem 30

Answer: A

Difficulty rating: 2270

Small Hint:

Big Hint:

Solution:

Problem 30 in Other Years