Points, Lines, and Planes

Basic Definitions

• Point: A location in space represented as a dot.
• Segment: Connects two points (e.g., Segment AB).
• Ray: Has a beginning but no end, one directional (e.g., Ray AB).
• Line: Extends infinitely in both directions, no beginning or end (e.g., Line AB).
• Plane: A flat, two-dimensional surface extending infinitely in the x and y directions.

Dimensions

• One-Dimensional: Segments, rays, and lines (travel in x direction for lines).
• Two-Dimensional: Planes (travel in x and y directions).

Collinear and Non-Collinear Points

• Collinear Points: Points that lie on the same line.
• Non-Collinear Points: Points that do not lie on the same line.

Coplanar and Non-Coplanar Points

• Coplanar Points: Points that lie on the same plane.
• Non-Coplanar Points: Points that do not lie on the same plane.

Determining Planes

1. Three Non-Collinear Points: Exactly one plane can pass through three non-collinear points.
2. A Line and a Point: Only one plane can pass through a line and a point.
3. Two Parallel Lines: Only one plane can pass through two parallel lines (coplanar lines).
4. Two Intersecting Lines: Intersecting lines lie on one plane (coplanar lines).

Non-Coplanar Lines

• Lines that do not share the same plane.
• Example: A line perpendicular to another line lying on a different plane.

Identifying Coplanar and Non-Coplanar Points/Lines

• Coplanar Points: Points lying on the same specified plane.
• Non-Coplanar Points: Combination of points from different planes.
• Coplanar Lines: Lines that exist on the same plane.
• Non-Coplanar Lines: Lines that don't share the same plane.

Example Problems

• Planes with Points:
  • Points E, D, C located on plane X.
  • Points F, D, R determine plane Y.
• Planes with Lines:
  • Line ED and Line AC determine plane X.
  • Line ED and Line FG determine plane Y.

Coplanar Segments

• Segment AB coplanar with Segment DC.
• Segment EF non-coplanar with AB and DC.

Verbal Questions and Identification

• Points on Planes:
  • Determine the plane for points A, D, M, F, R.
  • Intersection of planes X and Y is Line ED.
• Examples:
  • Identify coplanar points to ABCD on plane X: E, M, N.
  • Identify non-coplanar points to FB and G on plane Y: A, C, M, N.

Summary

Understanding the relationships between points, lines, and planes is crucial for solving geometric problems related to collinearity and coplanarity. These concepts are foundational in geometry.