 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Origamic II (Posted on 2012-03-18) 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 No Rating Comments: ( Back to comment list | You must be logged in to post comments.) Solution Comment 1 of 1
`Let a coordinate system be applied such thatthe 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 numbersgreater than zero and b > d > 0.`
`The creases EF anf JK coincide with theperpendicular bisectors of line segmentsAC 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 Please log in:
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