Question #5ab10
1 Answer
See explanation below.
Explanation:
We have to start from basic undefined objects of Geometry (point, straight line, plane) that satisfy basic axioms.
Then we will define a concept of half-plane and, based on this definition, it will be obvious that half-plane is not empty.
At this point it is important to understand the basic principles of Geometry, including undefined objects, relationship between them and axiom these objects must satisfy. A good introduction to this can be found on UNIZOR.COM in the chapter Geometry, topic Elements of Plane Geometry.
The Web gives not very rigorous explanation of half-plane:
A half-plane is a planar region consisting of all points on one side of an infinite straight line, and no points on the other side. If the points on the line are included, then it is called an closed half-plane.
We might define it a little more rigorously as follows.
Consider a plane
Consider a straight line
This straight line is a subset (
We also need a concept of a segment - a subset of a straight line in between two points on it. Any two points on a plane can be connected by a segment.
Consider all points on plane
Such an approach guarantees that our half-plane is non-empty since it contains, at least, one point
So, non-emptiness of half-plane is a direct consequence of its definition. More rigorous definition leads to non-emptiness relatively straight forward.