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Origamic II (Posted on 2012-03-18) Difficulty: 3 of 5
This is in continuation of Origamic.

A sheet of paper has the exact shape of a rectangle (denoted by ABCD) where AB is the longer side and AD is the shorter side. The vertex A is folded onto the vertex C, resulting in the crease EF (E on AB and F on CD).

The paper is thereafter unfolded and, the vertex A is folded onto F, resulting in the crease KJ.

Determine separately the ratio of the longer side (AB) of the rectangle to the shorter side (AD), whenever:

(i) K coincides with D and, J is on AE.

(ii) J coincides with B and, K is on AD.

(iii) J coincides with E and, K is on AD.

See The Solution Submitted by K Sengupta    
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Solution Solution Comment 1 of 1

Let a coordinate system be applied such that
the points have the following coordinates:
  A(0,0), B(b,0), C(b,d), D(0d), 
  E(e,0), F(fjd), J(j,0), and K(0,k)
where b, d, e, f, j, and k are real numbers
greater than zero and b > d > 0.
The creases EF anf JK coincide with the
perpendicular bisectors of line segments
AC an AF respectively.
 EF: y = (-b/d)x + q
     d/2 = (-b/d)(b/2) + q
     y = (-b/d)x + (b^2 + d^2)/(2d)
  E: 0 = (-b/d)e + (b^2 + d^2)/(2d)
     e = (b^2 + d^2)/(2b)
  F: d = (-b/d)f + (b^2 + d^2)/(2d)
     f = (b^2 - d^2)/(2b)
 JK: y = (-f/d)x + p
     d/2 = (-f/d)(f/2) + p
     y = (-f/d)x + (d^2 + f^2)/(2d)
  J: 0 = (-f/d)j + (d^2 + f^2)/(2d)
     j = (d^2 + f^2)/(2f)
       = (b^2 + d^2)^2/[4b(b^2 - d^2)]
  K: k = (-f/d)(0) + (d^2 + f^2)/(2d)
       = (d^2 + f^2)/(2d)
 --------------------------------------    
   i) K = D --> k = d 
            --> (d^2 + f^2)/(2d) = d
            --> d = f
            --> d = (b^2 - d^2)/(2b)
            --> 0 = b^2 - 2db - d^2
            --> b/d = 1 + sqrt(2)
                    ~= 2.4142
  ii) J = B --> j = b
            --> (b^2 + d^2)^2/[4b(b^2 - d^2)] = b
            --> 0 = 3b^4 - 6d^2b^2 = d^4
            --> b/d = sqrt(1 + 2*sqrt(3)/3)
                    ~= 1.4679
 iii) J = E --> j = e
            --> (b^2 + d^2)^2/[4b(b^2 - d^2)]
                  = (b^2 + d^2)/(2b)
            --> b/d = sqrt(3)
                    ~= 1.7321
 
 --------------------------------------
QED

 


  Posted by Bractals on 2012-03-18 13:24:13
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