Question

In geometry, the distance from a point to a line is defined to be the length of the perpendicular segment. What is the distance from a point to a segment defined as?

Answer 1

Definition:
The distance between two geometric objects is the shortest one among all possible distances between two points, one of which belongs to one object and another point - to another object.

Explanation:

Assume we have a segment `AB` and a point `P` not lying on a line `delta` that contains segment `AB`.

Drop a perpendicular from point `P` to line `delta`. Let point `Q in delta` be the base of this perpendicular. As we know, the length of segment `PQ` is the shortest one from point `P` to line `delta`, that is, it is shorter than the distance from `P` to any other (not `Q`) point on line `delta`..

There are three cases to consider.

Case 1:
`Q in AB`
Then `PQ`, as the shortest distance from point `P` to line `delta`, is the distance from point `P` to segment `AB`.

Case 2:
`Q notin AB`, but point `A` is closer to point `P` than point `B`.
Then `PA` is shorter than any other distance from point `P` to any point in `AB`. So, `PA` is the distance form point `P` to segment `AB`.

Case 3:
`Q notin AB`, but point `B` is closer to point `A` than point `B`.
Then `PB` is shorter than any other distance from point `P` to any point in `AB`. So, `PB` is the distance form point `P` to segment `AB`.