10.1 Points, Lines, and Planes

Geometry is all about shapes and spaces. From simple points to complex planes, it's the language we use to describe the world around us. Understanding these basics helps us make sense of everything from building designs to computer graphics.

Set theory and coordinate systems take geometry to the next level. They give us tools to analyze relationships between shapes and plot them precisely. This foundation is crucial for tackling more advanced geometric concepts and real-world applications.

Geometric Figures and Components

Components of geometric figures

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• Points represent a specific location in space denoted by a single uppercase letter (A, P) have no length, width, or depth
• Lines extend infinitely in both directions determined by two distinct points denoted by a lowercase letter or two uppercase points ([object Object],[object Object], [object Object],[object Object])
• Line segments consist of two endpoints and all the points between them have a finite length denoted by two uppercase letters with a line segment symbol above them ([object Object],[object Object])
• Rays consist of an endpoint and extend infinitely in one direction denoted by an endpoint and another point on the ray with a ray symbol above them ([object Object],[object Object])
• Planes are two-dimensional flat surfaces that extend infinitely in all directions determined by three non-collinear points or a line and a point not on the line denoted by a single uppercase script letter ([object Object],[object Object])
  • Planes are fundamental in geometry, which is the study of shapes, sizes, and positions of figures in space

Set Theory and Geometric Objects

Set theory in geometry

• Union of two or more geometric objects consists of all the points that belong to at least one of the objects denoted by the symbol [object Object],[object Object] ($\ell_1 \cup \ell_2$)
• Intersection of two or more geometric objects consists of all the points that are common to all the objects denoted by the symbol [object Object],[object Object] ($\ell_1 \cap \ell_2$)
  • Two lines can intersect at no point (parallel lines), one point, or infinitely many points (coincident lines)

Lines in the Cartesian Coordinate System

Parallel vs perpendicular lines

• Parallel lines are two lines in the same plane that do not intersect have the same slope vertical lines are parallel to each other and have undefined slope
• Perpendicular lines are two lines that intersect at a 90-degree angle slopes are negative reciprocals of each other horizontal and vertical lines are perpendicular to each other
• Cartesian coordinate system is a two-dimensional plane with a horizontal x-axis and a vertical y-axis origin is the point (0, 0) where the axes intersect used to graph and analyze geometric objects and their relationships
  • The Cartesian coordinate system is an example of a two-dimensional Euclidean space

Foundations of Geometry

• Geometry is built on a set of fundamental truths called axioms
• Axioms are statements accepted without proof and serve as the basis for deriving other geometric truths
• The concept of dimension is crucial in geometry, describing the number of coordinates needed to specify a point in a given space