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A rectangular sheet of paper, when halved by folding it at mid-point of its longer side, results in a rectangle, whose longer and shorter sides are in same proportion as the longer and shorter sides of the original rectangle. If the shorter side of the original rectangle is $2,$ what is the area of the smaller rectangle?

- $4 \sqrt{2}$
- $2 \sqrt{2}$
- $ \sqrt{2}$
- None of the above