Geometry Basics

Geometry

Definition
- Branch of mathematics concerned with shapes, sizes, properties of space, and the relative position of figures.
Basic Concepts
- Point: A location in space with no dimensions.
- Line: A straight one-dimensional figure having no thickness and extending infinitely in both directions.
- Plane: A flat two-dimensional surface that extends infinitely in all directions.
Types of Angles
- Acute Angle: Less than 90 degrees.
- Right Angle: Exactly 90 degrees.
- Obtuse Angle: Greater than 90 degrees but less than 180 degrees.
- Straight Angle: Exactly 180 degrees.
Triangles
- Types of Triangles:
  - Equilateral: All sides and angles are equal (60° each).
  - Isosceles: Two sides are equal, and the angles opposite those sides are equal.
  - Scalene: All sides and angles are different.
- Pythagorean Theorem: In a right triangle, ( a^2 + b^2 = c^2 ), where ( c ) is the hypotenuse.
Quadrilaterals
- Types:
  - Square: Four equal sides, four right angles.
  - Rectangle: Opposite sides equal, four right angles.
  - Rhombus: All sides equal, opposite angles equal.
  - Trapezoid (US) / Trapezium (UK): At least one pair of parallel sides.
Circles
- Radius: Distance from the center to any point on the circle.
- Diameter: Twice the radius, distance across the circle through the center.
- Circumference: The distance around the circle, calculated as ( C = \pi d ) or ( C = 2\pi r ).
- Area: The space contained within a circle, calculated as ( A = \pi r^2 ).
Solid Geometry
- 3D Shapes:
  - Cube: 6 equal square faces.
  - Rectangular Prism: 6 rectangular faces.
  - Sphere: All points equidistant from the center.
  - Cylinder: Two parallel circular bases connected by a curved surface.
  - Pyramid: A base that is a polygon and triangular faces that meet at a point.
Properties and Theorems
- Congruence: Two figures are congruent if they have the same shape and size.
- Similarity: Two figures are similar if they have the same shape but not necessarily the same size.
- Transversal: A line that cuts across two or more other lines, forming angles.
Coordinate Geometry
- Cartesian Coordinates: A system that uses pairs of numbers (x, y) to represent points on a plane.
- Distance Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
- Midpoint Formula: ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).
- Slope Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} ).
Applications of Geometry
- Used in fields such as architecture, engineering, art, and various sciences to model and analyze physical structures and spaces.

Geometry Overview

Geometry is a branch of mathematics focused on shapes, sizes, and the properties of space, including the relative positions of figures.

Basic Concepts

Point: Represents a specific location in space without dimensions.
Line: A straight figure that extends infinitely in both directions and has no thickness.
Plane: A flat surface that continues infinitely in all directions.

Types of Angles

Acute Angle: Measures less than 90 degrees.
Right Angle: Measures exactly 90 degrees.
Obtuse Angle: Measures more than 90 degrees but less than 180 degrees.
Straight Angle: Measures exactly 180 degrees.

Triangles

Equilateral Triangle: All sides and angles are equal, with each angle measuring 60 degrees.
Isosceles Triangle: Two sides are equal, and the angles opposite those sides are also equal.
Scalene Triangle: All sides and angles are different.
Pythagorean Theorem: In right triangles, the relationship ( a^2 + b^2 = c^2 ) holds, with ( c ) as the hypotenuse.

Quadrilaterals

Square: Consists of four equal sides with four right angles.
Rectangle: Features equal opposite sides and four right angles.
Rhombus: All four sides are equal, with opposite angles also equal.
Trapezoid (US) / Trapezium (UK): Defined by having at least one pair of parallel sides.

Circles

Radius: The distance from the center to any point on the circle.
Diameter: Twice the radius; it is the longest distance across the circle through the center.
Circumference: The distance around the circle, calculated using ( C = \pi d ) or ( C = 2\pi r ).
Area: The space contained within a circle, calculated by the formula ( A = \pi r^2 ).

Solid Geometry

Cube: A three-dimensional shape with six equal square faces.
Rectangular Prism: Composed of six rectangular faces.
Sphere: Defined by all points being equidistant from a central point.
Cylinder: Features two parallel circular bases and a curved surface connecting them.
Pyramid: Has a polygonal base and triangular faces that converge at a single point.

Properties and Theorems

Congruence: Figures are congruent if they share the same shape and size.
Similarity: Figures are similar if they share the same shape but not necessarily the same size.
Transversal: A line that intersects two or more other lines, creating various angles.

Coordinate Geometry

Cartesian Coordinates: Employs pairs of numbers (x, y) to represent points on a plane.
Distance Formula: Used to find the distance ( d ) between two points, expressed as ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
Midpoint Formula: Computes the midpoint ( M ) between two points, represented as ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).
Slope Formula: Determines the slope ( m ) of a line as ( m = \frac{y_2 - y_1}{x_2 - x_1} ).

Applications of Geometry

Geometry plays a crucial role in architecture, engineering, art, and various scientific fields, assisting in the modeling and analysis of physical structures and spaces.