Thread: "Find point C on the X-axis so that AC+BC is a minimum"

These are just thrown into an assignment on Reflections, and I for one don't get it.

We've got about 20 problems of them, so I can't really just try and hope it's right, either.

Examples:

A(1,4) B(4,2)

A(-2,3) B(6,3)

A(-3,2) B(-6,4)

A(1,-3) B(-3,-3)

I'm not even sure what it's asking for... Am I supposed to find a new point that would create the shortest possible line segments if graphed, or what?

If it's something else, what formulae would I use?

Hi

Yes you are supposed to find a new point that would create the shortest sum of the distances AC+BC, C being on X-axis

The solution is given in the sketch below
A' being the symmetric point of A with respect to X-axis
A'C=AC
AC+BC=A'C+BC
This latter distance is shortest when A',B and C are aligned

This problem is also known as "the problem of the river" : suppose that you are in A and you want to go to B after getting some water at the river (X-axis), where do you have to get the water in order to minimize the global distance ?