Question

A rectangular sheet of paper, when halved by folding it at the midpoint of its longer side, results in a rectangle, whose longer and shorter sides are in the 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?

• A. \(4\sqrt{2}\)
• B. \(2\sqrt{2}\)
• C. \(\sqrt{2}\)
• D. None of the above

Hint:

When dealing with proportionality problems, set up ratios for the sides of the rectangle.

Answer 1

The Correct Option is B

Let the longer side of the original rectangle be \(x\). After folding, the shorter side becomes the longer side of the smaller rectangle, and the longer side of the smaller rectangle is half the original longer side. Given that the proportions are the same, we have: \[ \frac{x}{2} = \frac{2}{x} \] Solving this gives \(x = 2\sqrt{2}\). Thus, the area of the smaller rectangle is \(2 \times \frac{2}{\sqrt{2}} = 2\sqrt{2}\). \[ \boxed{2\sqrt{2}} \]