Folded Rectangle, Find Width

The length of a rectangle is 5 inches and its width is less than 4 inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is √6, find the width of the rectangle.

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1. Draw CE. We show that G, E, C are collinear.
2. Consider △FGE and △CDE. FG = CD, ∠G ≅ ∠D, and GE = ED.
   ∴ △FGE ≅ △CDE and ∠E1 ≅ ∠E5. So G, E, C are collinear.
3. Consider △FBE and △CEB, B2 = E3 + E4, (FB || GC), BE = BE, and E2 = B3 (AC || FD).
   ∴ △FBE ≅ △CEB, so FE = 5 – y and ED = y.
4. Draw EH ⊥ ED. BH = 5 – 2y (HC = ED).
5. Consider △BHE.
   x² = (√6)² – (5 – 2y)² = 6 – (25 – 20y + 4y²)
   x² = 6 – 25 + 20y – 4y²
   and x² = (5 – y)² – y² (△FAB)
   x² = 25 – 10y + y² – y²
   So 25 – 10y = -19 + 20y – 4y²
   4y² – 30y + 44 = 0
   (2y – 4)(2y – 11) = 0
   y = 2 or 11/2. 11/2 > 4 n.g.
   So y = 2.
   x² = (5 – y)² – y² = 9 – 4 = 5;
   x = √5.