Question

One side of a rectangular piece of paper is $6\text{cm}$, the adjacent sides being longer than $6\text{cm}$. One corner of the paper is folded so that it sets on the opposite longer side. If the length of the crease is $l$ $\text{cm}$ and it makes an angle $\theta$ with the long side as shown, then $l$ is

• A. $\frac{3}{{\sin }^2\theta \cos \theta }$
• B. $\frac{6}{{\sin }^2\theta \cos \theta }$
• C. $\frac{3}{\sin \theta \cos \theta }$
• D. $\frac{3}{{\sin }^3\theta }$

Answer 1

The correct option is A $\frac{3}{{\sin }^2\theta \cos \theta }$

Let EC $=x$

In $\Delta BCE$,

$\frac{x}{l}=\sin \theta \Rightarrow x=l\sin \theta$

$EF=x=l\sin \theta$

In $\Delta ABF,$

$\angle ABF=90-2\theta$

$\angle AFB=2\theta$

$\sin 2\theta =\frac{6}{l\sin \theta }$

$\Rightarrow l=\frac{6}{\sin \theta .\sin 2\theta }$

$=\frac{6}{\sin \theta (2\sin \theta \cdot \cos \theta )}$

$l=\frac{3}{{\sin }^2\theta \cdot \cos \theta }$.