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Proportional Rectangular Sheet Folding

CAT 2004 Slot 1 · Quantitative Ability · Medium · Geometry

This is a medium Quantitative Ability question from the CAT 2004 Slot 1 paper. It tests Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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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 22, what is the area of the smaller rectangle?

  1. A.

    424\sqrt{2}

  2. B.

    222\sqrt{2}

  3. C.

    2\sqrt{2}

  4. D.

    None of the above

Answer

B

Explanation

Let the original rectangle have shorter side b=2b = 2 and longer side LL. When folded along the midpoint of its longer side, the new dimensions are shorter side L/2L/2 and longer side 22. Given proportions are equal: L2=2L/2  ⟹  L2=4L  ⟹  L2=8  ⟹  L=22\frac{L}{2} = \frac{2}{L/2} \implies \frac{L}{2} = \frac{4}{L} \implies L^2 = 8 \implies L = 2\sqrt{2} Area of the smaller rectangle =2×L2=L=22= 2 \times \frac{L}{2} = L = 2\sqrt{2}.

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