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Points, Lines, Planes, and Space Help (page 2)

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

Intersecting Line Principle

Suppose that lines L and M intersect in a point P . Then the two lines define a unique plane X . The following two statements are always true, as shown in Fig. 7-2:

  • An Expanded Set of Rules Points, Lines, Planes, and Space Intersecting Line Principle L and M lie in a common plane X
  • An Expanded Set of Rules Points, Lines, Planes, and Space Intersecting Line Principle X is the only plane in which both lines lie

 

An Expanded Set of Rules Points, Lines, Planes, and Space Intersecting Line Principle

Fig. 7-2 . Two lines L and M , intersecting at point P , define a specific plane X . The plane extends infinitely in 2D.

Line And Point Principle

Let L be a line and P be a point not on that line. Then line L and point P define a unique plane X . The following two statements are always true:

  • L and P lie in a common plane X
  •  X is the only plane in which both L and P lie

Plane Regions, Half Planes, and Intersecting Planes

Plane Regions

The 2D counterpart of the 1D line segment is the simple plane region . A simple plane region consists of all the points inside a polygon or enclosed curve. The polygon or curve itself might be included in the set of points comprising the simple plane region, but this is not necessarily the case. If the polygon or curve is included, the simple plane region is said to be closed . Some examples are denoted in Fig. 7-3A; the boundaries are drawn so they look continuous. If the polygon or curve is not included, the simple plane region is said to be open . In Fig. 7-3B, several examples of open simple plane regions are denoted; the boundaries are dashed.

 

An Expanded Set of Rules Points, Lines, Planes, and Space Plane Regions

Fig. 7-3 . Plane regions. At A, closed; at B, open.

The respective regions in Figs. 7-3A and 7-3B have identical shapes. They also have identical perimeters and identical interior areas. The outer boundaries do not add anything to the perimeter or the interior area.

These examples show specialized cases, in which the regions are contiguous, or “all of a piece,” and the boundaries are either closed all the way around or open all the way around. Some plane regions have boundaries that are closed part of the way around, or in segments; it is also possible to have plane regions composed of two or more non-contiguous sub-regions. Some such plane regions are so complex that they’re hard even to define. We won’t concern ourselves with such monstrosities, other than to acknowledge that they can exist.

Half Planes

Sometimes, mathematicians talk about the portion of a geometric plane that lies “on one side” of a certain line. In Fig. 7-4, imagine the union of all possible geometric rays that start at L , then pass through line M (which is parallel to L ), and extend onward past M forever in one direction. The region thus defined is known as a half plane .

 

 

An Expanded Set of Rules Points, Lines, Planes, and Space Half Planes

Fig. 7-4 . A half plane X , defined by two parallel lines, L and M . The half plane extends infinitely in 2D on the “M” side of L .

The half plane defined by L and M might include the end line L , in which case it is closed-ended . In this case, line L is drawn as a solid line, as shown in Fig. 7-4. But the end line might not comprise part of the half plane, in which case the half plane is open-ended . Then line L is drawn as a dashed line.

Parts of the end line might be included in the half plane and other parts not included. There are infinitely many situations of this kind. Such scenarios are illustrated by having some parts of L appear solid, and other parts dashed.

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