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Help finding a Theorem

  1. Nov 19, 2008 #1
    1. The problem statement, all variables and given/known data
    Let L be the segment of the Line y=3x+1 with end points (2, 7) and (6, 19).

    If P is a point not on L, is it guaranteed that the distance between P and L can be Minimized?


    2. Relevant equations

    None Really.


    3. The attempt at a solution

    I looked through my book but couldn't really find anything that was directly related to this. Is there some kind of famous theorem or something that should be obvious?
     
  2. jcsd
  3. Nov 19, 2008 #2
    Hi Cina,

    Are you familiar with the extreme value theorem in terms of compact sets? It says that a continuous, real-valued function defined on a compact set attains its extreme values.
     
  4. Nov 20, 2008 #3

    Mark44

    Staff: Mentor

    The slope of the given line is -1/3. Draw a line with this slope through each endpoint of the line segment. These two normal lines divide the plane into three regions.

    Case 1: If P lies between the two normals that were drawn, the minimum distance from the given line to P is the length along the normal from the given line to P.

    Case 2: If P lies outside of the two lines, the minimum distance from P to the line is the distance between P and the closer of the two endpoints.
     
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