Introduction to Geometry

Definition and Origin

• Geometry: Derived from Greek words geo (earth) and metry (measurement).
• General study of shapes, space, and their relationships.
• Encompasses lines, triangles, circles, angles, patterns, and 3D shapes.
• Visually mathematical field.

Basic Elements

Points

• Point: Basic unit, just a position with no dimensions.
  • Example: Points A, B, C, D.

Line Segments

• Line Segment: Set of connected points between two points (endpoints).
  • Example: Line segment AB.
• Endpoints: Starting and ending points of a line segment.
  • Example: Points A and B in line segment AB.
• Notation: Represented with endpoints and a line over them (e.g., (\overline{AB})).
• Length: Specified without a line on top (e.g., AB = 5 units).
• Dimensions: One-dimensional, as you can move back and forth along its direction only.

Rays

• Ray: Starts at a point (vertex) and extends indefinitely in one direction.
  • Example: Ray AD.
• Notation: Represented with an endpoint and an arrow (e.g., (\overrightarrow{AD})).
• Direction matters in notation ((\overrightarrow{AD}) ≠ (\overrightarrow{DA})).

Lines

• Line: Extends indefinitely in both directions.
  • Example: Line EF.
• Notation: Represented with arrows on both ends (e.g., (\overleftrightarrow{EF})).
• Infinite length and no endpoints.
• Collinear Points: Points that lie on the same line (e.g., X, Y, Z).

Midpoints

• Midpoint: Point exactly halfway between two other points on a line segment.
  • Example: If XZ = ZY, Z is the midpoint of line segment XY.

Dimensions

Zero Dimension

• Point: Position with no length, width, or height.

One Dimension

• Line Segment, Ray, Line: Objects with only length.

Two Dimensions

• Plane: Object with length and width but no height (e.g., the surface of a monitor).
• Planar: Any two-dimensional surface.

Three Dimensions

• 3D Space: Includes length, width, and height (e.g., moving in/out of the screen).

Advanced Concepts

• Higher-dimensional mathematics involves studying objects with more than three dimensions.