A piece of paper has the precise shape of a triangle (which we will denote by triangle ABC), where the respective lengths of the crease whenever the paper is folded such that; the vertex A is joined onto vertex B, the vertex A is joined onto vertex C and, the vertex B is joined onto vertex C are 7, 8 and 9.

Determine the lengths AB, AC and BC.

(In reply to re: Geometer's Sketchpad Solution by Jer)

I used the following property:

If XYZ is the triangle, then the crease length of folding 
vertex X to vertex Y is

  (|XY|/2) * tan( min( angle X , angle Y ) )

My first solution comes from A < C < B. Thus

  7 = crease length A -> B
    = (c/2)*tan(A)

  8 = crease length A -> C
    = (b/2)*tan(A)

therefore, c = (7/8)b

My second solution comes from B < C < A. Thus

  7 = crease length A -> B
    = (c/2)*tan(B)

  9 = crease length B -> C
    = (a/2)*tan(B)

therefore, c = (7/9)a

My third solution comes from C < B < A. Thus

  8 = crease length A -> C
    = (b/2)*tan(C)

  9 = crease length B -> C
    = (a/2)*tan(C)

therefore, b = (8/9)a

The only problem with the third solution
was that it cannot be constructed.

Posted by Bractals on 2012-05-22 01:53:27