Problem 27 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
foldingarea-decomposition
Zilda took a square sheet of paper with side 1 and made two folds, bringing two consecutive sides of the sheet onto a diagonal of it, as shown in the picture, obtaining a quadrilateral (the highlighted outline). What is the area of this quadrilateral?
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Answer: B — 2 − √2
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Hint 1 of 2
Set the square as a unit square and the diagonal from one corner; each fold brings a side onto that diagonal.
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Hint 2 of 2
Each crease bisects a 45° angle, hitting a side at distance tan 22.5° from a corner; find the kite's area.
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Approach: locate the creases and use the shoelace area
- With a unit square and diagonal from one corner, each fold creases along the bisector of a 45° angle.
- The two creases meet the far sides at distance tan 22.5° = √2 − 1 from the opposite corners.
- The remaining quadrilateral is a kite whose area computes to 2 − √2.
- The area is 2 − √2, choice B.
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