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 Another folded paper (Posted on 2012-01-03)

Fold corner A down to lie somewhere along side CD. Call x the portion of the way from C to D.

Corner B is outside of the original square and is now one side of a small triangle whose opposite side is along the remainder of side BC.

(A) Where would A be folded to make this small triangle isosceles?

(B) Find a formula in terms of x for the perimeter of this small triangle.

(C) Where would A be folded to maximize the area of this small triangle?

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`Let E and H be the points that cornersA and B are folded into respectivly.Let F and G be the intersections of thecrease with sides AD and BC respectively.Let I be the intersection of line EH with side BC. Let s be the side lengthof the square and x = |EC|.`
`Let beta = angle DAE. Then`
`                |DE|     |DC|-|EC|   tan(beta) = ------ = -----------                |AD|        |AD|`
`             = (s-x)/s.`
`Since F lies on FG (the perpendicularbisector of AE, |AF| = |FE|.`
`Let alpha = angle DFE. Looking attriangles DFE and AFE we see that`
`   alpha = 2*beta.`
`Therefore,`
`   cos(alpha) = cos(2*beta)`
`                 1 - tan(beta)^2              = -----------------                 1 + tan(beta)^2`
`                    x*(2s - x)              = ------------------      (1)                 2s^2 - 2sx + x^2`
`   sin(alpha) = sin(2*beta)`
`                   2*tan(beta)              = -----------------                 1 + tan(beta)^2`
`                    2s*(s - x)              = ------------------      (2)                 2s^2 - 2sx + x^2`
`Note that right triangles FDE, ECI, andGHI are similar. Thus,`
`   |EC| = x and |EI| = x/cos(alpha),`
`   |GH| = |GI|cos(alpha), and`
`   |HI| = |GI|sin(alpha).`
`To tie FDE and ECI together,`
`   |EI| + |IH| = |EH|`
`        or`
`   x/cos(alpha) + |GI|sin(alpha) = s`
`        or`
`             s*cos(alpha) - x   |GI| = -----------------------       (3)            cos(alpha)*sin(alpha)`
`-------------------------------------------`
`(A) If triangle GHI is isosceles, then`
`       cos(alpha) = sin(alpha)`
`                  or`
`       x*(2s - x) = 2s*(s - x)`
`                  or`
`       x^2 - 4sx + 2s^2 = 0`
`                  or`
`       x = s*[2 - sqrt(2)].`
`-------------------------------------------`
`(B) Perimeter(GHI) = |GH| + |HI| + |IG|`
`    Combining this with (1)-(3) gives`
`    Perimeter(GHI) = x.`
`-------------------------------------------`
`(C) Maximize Area(GHI):`
`    Area(GHI) = |GH|*|HI|/2`
`                 |GI|^2*cos(alpha)*sin(alpha)              = ------------------------------                               2`
`    Combining this with (1)-(3) gives`
`                 sx^3 - x^4    Area(GHI) = ------------            (4)                 8s^2 - 4sx`
`    Maximum area when`
`      (8s^2 - 4sx)*d/dx(sx^3 - x^4) -`
`      (sx^3 - x^4)*d/dx(8s^2 - 4sx) = 0`
`             or`
`      (8s^2 - 4sx)*(3sx^2 - 4x^3) -`
`      (sx^3 - x^4)*(-4s) = 0`
`             or`
`      4sx^2*(3x^2 - 10sx +6s^2) = 0 `
`             or`
`      x = s*[5 - sqrt(7)]/3             (5)          Combining (4) and (5) gives`
`                      s^2*[316 - 119*sqrt(7)]    Max. Area(GHI) = -------------------------                               54`
`QED-------------------------------------------`
`Note the above results checked withGeometer's Sketchpad.`
` `

 Posted by Bractals on 2012-01-03 13:23:18

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