# Math Help - Line Segment Distance

1. ## 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."

2. Originally Posted by furnis1
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?

3. Originally Posted by furnis1

"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.

Originally Posted by furnis1

Locate a point 'X' on 'M' where the difference between |AX| and |BX| is a maximum."
Restate your argument to clear these questions up.

4. 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 $A-C-B$.

5. Originally Posted by furnis1
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.

6. Originally Posted by masters
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)

7. Originally Posted by Plato
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

8. Originally Posted by furnis1
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.

9. Originally Posted by furnis1
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.

10. ## 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.

11. ## Looking between the lines

Originally Posted by Mathstud28
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