Enterprise

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

 

 
 
 
 
 
 
 
 
 
 
 
 

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i>Created 22 March 2006, Last Update10 June 2020

Why It Works

Extend the folded center line and construct a line parallel to the tilted end of the paper. The two colored triangles are obviously similar, and we have two equations in two unknowns.

  1. (1-A)/c = (1-c)/a
  2.  A(1-A) = C(1-c)
  3. A - A2 = C - C2
  4. A - C  = A2 - C2 = 1/4
  5. C = A + 1/4
  6. A - A2 = C - C2 = A + 1/4 - (A + 1/4)2
  7. A - A2 = A + 1/4 - A2 - 2(1/4)A - 1/16
  8. 0 = 1/4 - A/2 -1/16
  9. A/2 = 3/16 or A = 3/8
  10. C = A + 1/4 = 5/8

So A = 3/8, the second side = 1/2 = 4/8 and C = 5/8. The triangle is a 3-4-5 right triangle. Furthermore, the blue triangle has edges 1-A = 5/8 and 1-c= 3/8, and must be congruent to the yellow triangle.

The magenta angles are arctan 0.75 = 36.86 degrees, not a bad approximation to 180/5.