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Rectangle-Diagonal (Posted on 2016-11-02) Difficulty: 3 of 5

  
Let E be the foot of the perpendicular from A to diagonal
BD in rectangle ABCD. Let F and G be points on sides BC
and CD respectively such that EFCG is a rectangle.

If |BD| = d, |EG| = g, and |EF| = 1, find d in terms of g.
  

  Submitted by Bractals    
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Solution: (Hide)

  
Let |EB| = b and |EF| = f.

Pythagoras' theorem applied to ΔBFE gives b2 = f2 + 1.

ΔAEB ~ ΔBFE ⇒ |AB|/b = |AB|/|BE| = |BE|/|EF| = b/1
                          ⇒ |AB| = b2.

ΔDAB ~ ΔBFE ⇒ d/b2 = |BD|/|AD| = |BE|/|EF| = b/1
                          ⇒ d = b3 ⇒ b2 = d2/3.

|DG| = |CD| - |CG| = |AB| - |EF| = b2 - 1 = f2.

ΔEGB ~ ΔBFE ⇒ g/f2 = |EG|/|DG| = |BF|/|EF| = f/1
                          ⇒ g = f3 ⇒ f2 = g2/3.

b2 = f2 + 1 ⇒ d2/3 = g2/3 + 1 ⇒ d = ( g2/3 + 1 ) 3/2.

QED
  

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  Subject Author Date
SolutionJer2016-11-02 14:26:28
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