Overview
This lecture covers basic geometry concepts including points, lines, planes, collinear and coplanar points, methods for determining planes, and examples distinguishing between coplanar and non-coplanar elements.
Points, Lines, Segments, and Rays
- A point is a specific location in space, represented by a dot.
- A segment connects two points and has two endpoints.
- A ray starts at one point and extends infinitely in one direction.
- A line has no endpoints and extends infinitely in both directions.
- Segments, rays, and lines are one-dimensional.
Planes and Their Properties
- A plane is a flat, two-dimensional surface extending infinitely in x and y directions.
- Planes can contain points, lines, segments, and rays.
Collinear and Non-Collinear Points
- Collinear points lie on the same straight line.
- Non-collinear points cannot all be connected by one line.
Coplanar and Non-Coplanar Points/Lines/Segments
- Coplanar points or lines exist on the same plane.
- Non-coplanar points or lines do not all share the same plane.
- Coplanar segments exist on the same plane; non-coplanar segments do not.
Determining a Plane
- Three non-collinear points determine a unique plane.
- A line and a point not on the line determine a unique plane.
- Two parallel lines determine a unique plane.
- Two intersecting lines determine a unique plane.
Examples & Q&A
- Points a, b, and c are collinear if on the same line; otherwise, they are non-collinear.
- Four points a, b, c, and d are coplanar if all lie on the same plane.
- Points a, b, c, d, and g are coplanar if in the same plane; adding e or f can make them non-coplanar.
- Lines ab and cd are coplanar if on the same plane; adding a non-coplanar line like ef changes this.
- Segments ab and dc are coplanar; segment ef is non-coplanar to these.
- Points d, e, and c all found in plane x, so they determine plane x.
- Intersection of two planes (x and y) is a line, such as line ed.
Key Terms & Definitions
- Point — a specific location in space.
- Line — a straight path extending infinitely in both directions.
- Segment — part of a line with two endpoints.
- Ray — starts at one point, extends infinitely in one direction.
- Plane — a flat, two-dimensional surface extending infinitely.
- Collinear — points that lie on the same line.
- Coplanar — points or lines that lie on the same plane.
- Non-collinear — points not all on the same line.
- Non-coplanar — points or lines not all on the same plane.
Action Items / Next Steps
- Practice identifying collinear, coplanar, non-collinear, and non-coplanar points in diagrams.
- Review definitions and methods for determining planes.
- Prepare for questions involving planes, lines, and points for homework or a quiz.