Euclidean Geometry

Geometry is a branch of mathematics.

• It deals with the properties and relations of geometric objects.
• These objects include points, lines, surfaces, and solids.
• Higher dimensional analogs are also a concern.
• Geometry is one of the oldest mathematical sciences.
• It emerged independently in many early cultures.
• It was useful for dealing with measurements like lengths, areas, and volumes.

Euclidean Geometry

• Euclidean geometry studies geometrical shapes.
• These shapes can be plane or solid.
• This study is based on axioms and theorems.
• It is named after Euclid, a Greek mathematician.
• Euclid compiled and systematized it in "Elements".
• Key concepts are points, lines, planes, angles, and shapes.
• Point, line, and plane are fundamental elements.
• A point is a location lacking dimension.
• A line is straight and one-dimensional with no thickness.
• Lines extend infinitely in both directions.
• A plane is a flat, two-dimensional surface.
• Planes extend infinitely.
• Angles are formed by two rays sharing a vertex.
• Shapes include triangles, squares, circles (2D), cubes, spheres, and pyramids (3D).

Axioms and Postulates of Euclidean Geometry

• A straight line can join any two points.
• A straight line can be extended to any finite length.
• A circle can be described with any center and distance.
• All right angles are equal.
• If a line intersects two lines, making interior angles on one side less than two right angles, the lines meet on that side if extended indefinitely.

Basic Geometric Shapes

• Triangle: A polygon with three edges and three vertices.
  • Types: Equilateral, isosceles, scalene, right.
• Square: A quadrilateral with four equal sides and four right angles.
• Rectangle: A quadrilateral with four right angles.
• Circle: The set of all points in a plane equidistant from a center point.
• Polygon: A closed 2D shape with straight line edges.
  • Examples: Pentagon, hexagon, octagon.

Angles

• Acute Angle: Measures less than 90 degrees.
• Right Angle: Measures exactly 90 degrees.
• Obtuse Angle: Measures greater than 90 and less than 180 degrees.
• Straight Angle: Measures exactly 180 degrees.
• Reflex Angle: Measures greater than 180 and less than 360 degrees.
• Complementary Angles: Two angles sum to 90 degrees.
• Supplementary Angles: Two angles sum to 180 degrees.

Theorems in Geometry

• Pythagorean Theorem: In a right triangle, a² + b² = c², where c is the hypotenuse.
• Angle Sum of a Triangle: Interior angles of a triangle always sum to 180 degrees.
• Vertical Angles Theorem: Vertical angles are congruent.

Coordinate Geometry

• Coordinate geometry uses a coordinate system to represent points, lines and shapes.
• The Cartesian coordinate system uses two perpendicular axes (x and y)
• It defines the position of a point in a plane using ordered pairs (x, y).
• Distance Formula: Distance between (x₁, y₁) and (x₂, y₂) is √((x₂ - x₁)² + (y₂ - y₁)²).
• Midpoint Formula: Midpoint of the line segment joining (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Slope of a Line: Slope (m) of a line through (x₁, y₁) and (x₂, y₂) is (y₂ - y₁) / (x₂ - x₁).
• Equation of a Line:
  • Slope-intercept form: y = mx + b (m is slope, b is y-intercept).
  • Point-slope form: y - y₁ = m(x - x₁) (m is slope, (x₁, y₁) is a point on the line).
  • Standard form: Ax + By = C.

Three-Dimensional Geometry

• Three-dimensional geometry extends 2D concepts to three dimensions.
• It uses three coordinate axes (x, y, z) to define points in space.
• Points are defined using ordered triples (x, y, z).
• Key concepts include:
  • Distance Formula in 3D: Distance between (x₁, y₁, z₁) and (x₂, y₂, z₂) is √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
  • Equations of planes and lines in 3D space.
• Basic 3D Shapes:
  • Sphere: The set of all points in space equidistant from a center point.
  • Cube: A solid with six square faces.
  • Cylinder: A solid with two parallel circular bases connected by a curved surface.
  • Cone: A solid with a circular base and a single vertex.

Transformations

• Transformations are operations that change a geometric figure's position, size, or orientation.
• Types of Transformations:
  • Translation: Moving a figure without changing its size or orientation.
  • Rotation: Turning a figure around a fixed point.
  • Reflection: Creating a mirror image of a figure across a line.
  • Dilation: Changing the size of a figure by a scale factor.

Congruence and Similarity

• Congruent Figures: Figures that have the same shape and size.
  • Corresponding sides and angles are equal.
• Similar Figures: Figures that have the same shape but different sizes.
  • Corresponding angles are equal, and corresponding sides are proportional.

Area and Volume

• Area: Measures the 2D space inside a closed figure.
  • Formulas for basic shapes:
    • Triangle: (1/2) * base * height
    • Square: side²
    • Rectangle: length * width
    • Circle: π * radius²
• Volume: Measures the 3D space inside a solid.
  • Formulas for basic solids:
    • Cube: side³
    • Sphere: (4/3) * π * radius³
    • Cylinder: π * radius² * height
    • Cone: (1/3) * π * radius² * height

Non-Euclidean Geometry

• Modifies or challenges Euclid's postulates, especially the parallel postulate.
• Hyperbolic and elliptic geometry are examples.
• Hyperbolic Geometry: Features multiple lines parallel to a given line through a point not on the line.
• Elliptic Geometry: Features no parallel lines.
• It is used in fields like astronomy and general relativity.