Can the distance between a point and a line segment be minimized?

[CINA]

Homework Statement

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?

Homework Equations

None Really.

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?

 

[Unco]

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.

 

[Mark44]

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.