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Math Help - Line Segment Distance

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    Line Segment Distance

    Hey guys. I am having extreme dificulty on this question. I am wondering if anyone knows how to go about this, seems some what too difficult for me:

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'. Locate a point 'X' on 'M' where the difference betwween |AX| and |BX| is a maximum."
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    Quote Originally Posted by furnis1 View Post
    Hey guys. I am having extreme dificulty on this question. I am wondering if anyone knows how to go about this, seems some what too difficult for me:

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'. Locate a point 'X' on 'M' where the difference betwween |AX| and |BX| is a maximum."
    The midpoint?
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  3. #3
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    Quote Originally Posted by furnis1 View Post

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'.
    What do you mean "opposite sides"? A straight line only has length. Do you mean above and below the line? Or, are 'A' and 'B' endpoints of a line segment? Also, it is conventional in geometry to use lower case letters when designating a line with a single letter.


    Quote Originally Posted by furnis1 View Post

    Locate a point 'X' on 'M' where the difference between |AX| and |BX| is a maximum."
    Restate your argument to clear these questions up.
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    This is to answer the question and thetwo responses.
    Assume we are operating in a plane. Modern axiomatic geometry has axioms about sides of a line.
    The axiom vary but go something like this.
    Definition: Two points are on different sides of l if and only if there is a point of l between the points.

    Given a line l there is a point P such P \notin l.
    The is at least Q \in l .
    There is a point R such that P-Q-R ( Q is between P\,\&  Q \,).
    Now we have a P-\mbox{side of }l and R-\mbox{side of }l.

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'.
    Definition there is a point C \in M such that A-C-B.
    Last edited by Plato; July 15th 2008 at 03:23 PM.
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    Quote Originally Posted by furnis1 View Post
    Hey guys. I am having extreme dificulty on this question. I am wondering if anyone knows how to go about this, seems some what too difficult for me:

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'. Locate a point 'X' on 'M' where the difference betwween |AX| and |BX| is a maximum."
    Say AX is longer than XB.

    I also don't know how to solve it because all I can see is AX +XB = M.

    By "common sense", the difference between AX and XB will be maximum when XB is minimum.
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    Quote Originally Posted by masters View Post
    What do you mean "opposite sides"? A straight line only has length. Do you mean above and below the line? Or, are 'A' and 'B' endpoints of a line segment? Also, it is conventional in geometry to use lower case letters when designating a line with a single letter.




    Restate your argument to clear these questions up.
    A and B are on opposite sides of the line (above and below)....line segment 'm' (if lower case helps)
    Last edited by furnis1; July 15th 2008 at 03:52 PM.
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    Quote Originally Posted by Plato View Post
    This is to answer the question and thetwo responses.
    Assume we are operating in a plane. Modern axiomatic geometry has axioms about sides of a line.
    The axiom vary but go something like this.
    Definition: Two points are on different sides of l if and only if there is a point of l between the points.

    Given a line l there is a point P such P \notin l.
    The is at least Q \in l .
    There is a point R such that P-Q-R ( Q is between P\,\& Q \,).
    Now we have a P-\mbox{side of }l and R-\mbox{side of }l.

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'.
    Definition there is a point C \in M such that A-C-B.

    thanks for an answer but i was wondering if you can clear it up for a me a bit further. I couldnt understand it properly. Thanks again
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  8. #8
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    Quote Originally Posted by furnis1 View Post
    Hey guys. I am having extreme dificulty on this question. I am wondering if anyone knows how to go about this, seems some what too difficult for me:

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'. Locate a point 'X' on 'M' where the difference betwween |AX| and |BX| is a maximum."
    I've made a sketch of the situation.

    If the difference of distances should be a minimum then the points A, B and X form an isoscle triangle.

    Without any further informations about the line m and the points A and B it is impossible for me to answer your question.
    Attached Thumbnails Attached Thumbnails Line Segment Distance-max_distance.gif  
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    Quote Originally Posted by furnis1 View Post
    Hey guys. I am having extreme dificulty on this question. I am wondering if anyone knows how to go about this, seems some what too difficult for me:

    "IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'. Locate a point 'X' on 'M' where the difference betwween |AX| and |BX| is a maximum."
    I've played a little bit with your problem.
    I've noticed that the maximum difference seems to occur if the line m is the angle bisector of the angle \angle(BXA). But unfortunately I haven't found a proof for this statement .

    If the point X is approaching the "ends" of the line the difference of distances has a limit which is represented by the green line segment.
    Attached Thumbnails Attached Thumbnails Line Segment Distance-max_distan2.gif  
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  10. #10
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    how to find point X

    You are asked to find the point X - and I hope nobody asked you to prove that this point is the only one satisfying all conditions

    1. draw the perpendicular bisector of AB which cuts m in M.
    2. the difference of distances AM - BM = 0 therefore the differences have a minimum if X is placed on M
    3. draw the point A' which is the reflection of A over m.
    4. draw the line BA' which cuts m in X. And that's exactly(?) the point you are looking for.
    Attached Thumbnails Attached Thumbnails Line Segment Distance-max_distan2konstrkt.gif  
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  11. #11
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    Looking between the lines

    Quote Originally Posted by Mathstud28 View Post
    The midpoint?
    It seems to me what you are trying to solve is AB = AP + PB in my own terms. You should read Geometry for dummies has all the answers in

    David
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