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Night of the living straight edge!

  1. Apr 14, 2005 #1
    This one's just cute, but it's pretty easy (did it on a napkin at McDonald's when I was bored :uhh: ). We all know that given a line segment, we can easily construct a perpendicular bisector with a straight edge and a compass. The challenge is to construct a perpendicular bisector to a given line segment of unknown length using only a straight edge, in a finite amount of steps (and a pencil, of course). The straight edge is a rectangle, but cannot be used to measure the length of the line segment.
     
  2. jcsd
  3. Apr 14, 2005 #2
    My answer:
    Place the rectangle along the line so that the edge of the rectangle is at the terminating point. Draw a line perpendicular to the segment. Do the same for the other side, and close the resulting rectangle. Draw the two diagonals. They will intersect at the halfway point of the line segment. Repeat this again on the other side of the line so that you have two rectangles of equal length. The two sets of diagonals are at the halfway point along the axis of the line segment, so connecting the two intersection points of the diagonals will bisect the line exactly.

    Correct?
     
  4. Apr 15, 2005 #3
    Yep

    That'll do it! There's another way as well, without having to have a right-angle present initially. For those who are still reading, restrict the problem as stated further in this manner: The straight edge is "infinitely long", so you cannot access all four sides, only two parallel sides. It is of adjustable width. You can rotate the straight edge, but of course you cannot tell from sight whether you have rotated it exactly 90 degrees.
     
  5. Apr 15, 2005 #4
    Fold the napkin so that the endpoints of the line meet. draw a line along the crease with the straight edge.

    I'm not sure if you mean to use the 'rectangular' nature of the stright edge or not. In classic straight edge and compass geometry this is illegal.
     
  6. Apr 15, 2005 #5
    I knew I was forgetting something. Since I didn't forbid it, it's a valid answer. :smile: An additional restriction is that you cannot fold the surface that the line segment is on.
    Also, the reason I allow the straight edge to be 2-dimensional as opposed to its stricter cousin is in order to compensate for something the compass would normally be doing.
     
    Last edited: Apr 15, 2005
  7. Apr 15, 2005 #6
    Set the width of the straightedge to some width less than the length of the line segment. Position it so that one edge of the straightedge passes over one endpoint of the line segment, and the opposite edge of the straightedge passes over the other endpoint of the line segment. Now trace both sides of the straightedge. Now reverse it, so if endpoint A was with straightedge side 1 and endpoint B was with side 2, A is now with 2 and B is now with 1, and again trace both sides of the straightedge. You now have made a diamond shape with the line segment connecting two corners of the diamond. Draw a line passing through the other two corners of the diamond and this will be the perpendicular bisector to the line segment.
     
  8. Apr 15, 2005 #7
    Good job! :smile: Curiously, you cannot create a perpendicular if you just stick to plane rotations.
     
  9. Apr 16, 2005 #8
    How do you mean? BicycleTree's solution works of course just as well when you rotate the straight edge, like this:
     

    Attached Files:

  10. Apr 20, 2005 #9
    Simple if I have a rectangle with diagonal longer than the straight line. Say the rectangle is designated by ABCD and the line by EF. Coincide A and E. Let BC be cut any where by F and mark B. Do it on the other side of the line and mark B'. BB' bisects the original line. (PS: Two lines originating from two edges of diameter form a right angle triangle on the circumference.

    I got the clue of reading hidden posts after submitting my reply. One of the guys already answered it.
     
    Last edited: Apr 20, 2005
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