HMMT 二月 2008 · 几何 · 第 5 题

英文解析

5. [ 5 ] A piece of paper is folded in half. A second fold is made such that the angle marked below has
   ° °
   measure φ (0 < φ < 90 ), and a cut is made as shown below.
   When the piece of paper is unfolded, the resulting hole is a polygon. Let O be one of its vertices.Á
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   Suppose that all the other vertices of the hole lie on a circle centered at O , and also that ∠ XOY = 144 ,
   where X and Y are the the vertices of the hole adjacent to O . Find the value(s) of φ (in degrees).
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   Answer: 81 Try actually folding a piece of paper. We see that the cut out area is a kite, as shownbelow. The fold was made on AC , and then BE and DE . Since DC was folded onto DA , we have
   ∠ ADE = ∠ CDE .
   A CEB
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   Either A or C is the center of the circle. If it’s A , then ∠ BAD = 144 , so ∠ CAD = 72 . Using
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   CA = DA , we see that ∠ ACD = ∠ ADC = 54 . So ∠ EDA = 27 , and thus φ = 72 + 27 = 99 ,
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   which is inadmissible, as φ < 90 .
   ° ° ° ° °
   So C is the center of the circle. Then, ∠ CAD = ∠ CDA = 54 , ∠ ADE = 27 , and φ = 54 +27 = 81 .
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