Find the point of the line

L

Lolyta

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Feb 27, 2012
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15
Hello.
If we have two points A,B and a line (both points are in the same side of the line) how could I find the point Q of the line which minimizes the sum: distance(A,Q)+distance(Q,B).

Thanks for every suggestion!
 
Last edited:
Lolyta said:
two ... lines s and t and [points A and Q]
Click to expand...

I'm thinking that this depends upon where the lines and points are located, all in relation to one another.

Are you able to upload a diagram?
 
I started to write about "setting up a coordinate system" but realized that is not a very good way of doing this problem.


Suppose Q is anywhere between A and B. Do you see that sum of the two distances is equal to the distance between A and B?

Suppose Q is not between A and B. Do you see that one of the distance is larger than the distance between A and B and the other is positive?

(The point of this is that the answer is not a single point!)
 
Hello, Lolyta!

If we have two points \(\displaystyle A,B\) and a line (both points are in the same side of the line),
how could I find the point \(\displaystyle Q\) on the line which minimizes the sum: .\(\displaystyle \overline{AQ} + \overline{QB}\,?\)
Click to expand...

Code: 
        A
        *
                  B
                  *
                  |
                  |
                  |
    - - - - - - - * - -
                  |P
                  |
                  |
                  *
                  B'
Drop a perpendicular \(\displaystyle BP\) from point \(\displaystyle B\) to the line.
Extend the perpendicular to point \(\displaystyle B'\) so that \(\displaystyle BP = PB'.\)

Code: 
        A
        *
         \        B
          \       *
           \     /|
            \   / |
             \ /  |
    - - - - - * - + - -
              Q\  |
                \ |
                 \|
                  *
                  B'
Draw \(\displaystyle AB'\) intersecting the line at \(\displaystyle Q.\)

\(\displaystyle Q\) is the desired point.
 
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