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Math Help - Geometry constructions

  1. #1
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    Geometry constructions

    I don't know how to construct the following:
    "Draw a line segment so close to the edge of your paper that you can swing arcs on only one side of the segment. Then construct the perpendicular bisector of the segment."
    I'm not sure what they're asking for. How is this construction supposed to look like?
    Thanks!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by jmmsan View Post
    I don't know how to construct the following:
    "Draw a line segment so close to the edge of your paper that you can swing arcs on only one side of the segment. Then construct the perpendicular bisector of the segment."
    I'm not sure what they're asking for. How is this construction supposed to look like?
    Thanks!
    it looks like this..
    the thicker lines are the edges of the paper..
    you will draw the segment AC somewhere near the edge, then you will draw line BD such that |AB| = |BC| and BD is perpendicular to AC..
    the reason why they asked you to draw it near the edge is because you will be using compass to be able to draw the perpendicular bisector..
    Attached Thumbnails Attached Thumbnails Geometry constructions-sketch.jpg  
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  3. #3
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    I recall the usual, (easiest?), way to construct the perpendicular bisector of a line segment is to draw circular arcs of the same radius centered on each end of the line segment. The radius is longer than the approximate half-length of the line. The circular arcs intersect on both sides of the line. The perpendicular bisector of the line segment then is the line connecting the two intersection points.

    In your Problem now, the designer/thinker of the Problem made sure you cannot do that. He eliminated one side of the line segment for the other intersection point of the two circular arcs.

    If measuring, or using a tape measure is allowed in your "construction", then draw the usual circular arcs of the same radius that is longer than the approximate half-length of the line segment. You have only one intersection point.
    Then, by tape measure, mark the midpoint pof the line segment. Connect this midpoint to the arcs intersection point and that is the perpendicular bisector of the line segment.

    If measuring is not allowed, then get another intersection point on the same side of the line segment. This time, use a radius not equal to the first radius but still longer than the approximate half-length of the line segment. This new radius may be longer or shorter than the first radius.
    The perpendicular bisector of the line segment then is the line passing through the two intersection points and the line segment too.
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  4. #4
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    Quote Originally Posted by jmmsan View Post
    I don't know how to construct the following:
    "Draw a line segment so close to the edge of your paper that you can swing arcs on only one side of the segment. Then construct the perpendicular bisector of the segment."
    I'm not sure what they're asking for. How is this construction supposed to look like?
    Thanks!
    Hello,

    let the line segment be AB (A and B are the endpoints of the segment), let l be the length of AB.

    1. draw an arc around A with r_1 > \frac12 \cdot l and with the same radius draw an arc around B. These 2 arcs intersect in one point.

    2. draw an arc around A with r_2 > r_1 and with the same radius draw an arc around B. These 2 arcs intersect in one point.

    3. draw a line through these 2 intersection points. This line is the perpendicular bisector of AB.
    Attached Thumbnails Attached Thumbnails Geometry constructions-mitsenkr_doppelkrs.gif  
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