Geometry: Sets and Points Terminology

Set Theory

• Set: A collection of elements enclosed in curly braces { }. Elements can be numbers, letters, symbols, etc.

Types of Sets

• Finite Set:

  • Definition: A set that contains a countable number of elements.
  • Example: {12, 25, 33, 40}

• Infinite Set:

  • Definition: A set that contains an uncountable number of elements.
  • Example: {5, 10, 15, 20,…}

• Empty Set:

  • Definition: A set with no elements.
  • Shown as: { } or Ø

Basic Geometric Concepts

Point

• Definition: A location in space, represented by a dot and named by a capital letter.
  • Properties: It has NO length, width, or thickness.
  • Symbol: A. - Indicates a place or position

Line

• Definition: A set of infinite points that extends endlessly in both directions.
  • Representation:
  • Can be named in different ways:
    • Using two capital letters: AB, AC
    • One lowercase cursive letter: l, t, a
  • To define a line, at least two points are needed.

Plane

• Definition: A set of points that form a completely flat surface extending indefinitely in all directions.
  • Properties: A plane has no boundaries.
  • Naming:
  • Can be named in different ways:
    • Using three capital letters: ABL
    • Using one lowercase cursive letter: c

Collinear and Noncollinear Points

Collinear Points

• Definition: Three or more points that lie on the same line.
  • Example: Points A, B, C are collinear points.

Noncollinear Points

• Definition: Three or more points that do NOT lie on the same line.
  • Example: Points J, M, L are noncollinear points.

Line Segment

• Definition: A subset or part of a line, consisting of two points on a line, called endpoints.
  • Naming: A line segment can be named using its endpoints, such as AB or BA.

Midpoint

• Definition: The point on a line segment that divides the segment into two congruent (equal) parts.

  • Notation: The segments can be labeled such that if AB is the line segment, then:
  • AM + MB = AB
  • 2AM = AB => AM = MB
  • Symbol: m is the midpoint.

• Key Point: A line does NOT have a midpoint; only a line SEGMENT can have a midpoint.

Congruent Segments

• Definition: Line segments that have equal measure.

Line Bisector

• Definition: Any line or part of a line that intersects a line segment at its midpoint.
  • The line bisector divides the segment into two equal halves, thereby creating congruent segments.