Geometry Basics

Geometry

• Definition: Study of shapes, sizes, and properties of space.

• Basic Concepts:

  • Point: An exact location in space, no dimensions.
  • Line: A straight one-dimensional figure with length, extending infinitely in both directions.
  • Plane: A flat two-dimensional surface extending infinitely.

• Types of Angles:

  • Acute: Less than 90 degrees
  • Right: Exactly 90 degrees
  • Obtuse: Greater than 90 degrees but less than 180 degrees
  • Straight: Exactly 180 degrees

• Shapes:

  • Triangles: Sum of angles = 180 degrees.
    • Types: Equilateral, Isosceles, Scalene.
  • Quadrilaterals: Sum of angles = 360 degrees.
    • Types: Square, Rectangle, Parallelogram, Trapezoid.
  • Circles: Defined by radius, diameter, and circumference.
    • Area = πr², Circumference = 2πr.

• Theorems:

  • Pythagorean Theorem: In a right triangle, a² + b² = c².
  • Congruence: Two shapes are congruent if they have the same size and shape.

Algebra

• Definition: Branch of mathematics dealing with symbols and rules for manipulating those symbols.

• Basic Concepts:

  • Variables: Symbols that represent unknown values (e.g., x, y).
  • Expressions: Combinations of variables, numbers, and operations (e.g., 3x + 2).
  • Equations: Statements of equality (e.g., 2x + 3 = 7).

• Operations:

  • Addition, Subtraction, Multiplication, and Division: Fundamental arithmetic operations applied to algebraic expressions.

• Solving Equations:

  • Linear Equations: In the form ax + b = c.
  • Quadratic Equations: In the form ax² + bx + c = 0; solutions found using factoring, completing the square, or the quadratic formula.

• Functions:

  • Definition: A relation between a set of inputs and a set of possible outputs, often expressed as f(x).
  • Types:
    • Linear Functions: f(x) = mx + b; graph is a straight line.
    • Quadratic Functions: f(x) = ax² + bx + c; graph is a parabola.

• Inequalities:

  • Represent a range of values rather than a single value (e.g., x > 5).
  • Can be solved similarly to equations, but the direction of the inequality can change when multiplying/dividing by a negative number.

• Polynomials:

  • Expressions involving variables raised to whole number powers (e.g., 2x³ + 3x² - x + 5).
  • Can be added, subtracted, multiplied, and divided.

Geometry

• Definition: Study of shapes, sizes, and properties of space.
• Basic Concepts:
  • Point: A specific location with no dimensions.
  • Line: Straight, one-dimensional figure extending infinitely in both directions.
  • Plane: Flat, two-dimensional surface extending infinitely.
• Types of Angles:
  • Acute: Less than 90 degrees.
  • Right: Exactly 90 degrees.
  • Obtuse: Greater than 90 degrees but less than 180 degrees.
  • Straight: Exactly 180 degrees.
• Shapes:
  • Triangles:
    • Sum of angles = 180 degrees.
    • Types:
      • Equilateral: All sides equal.
      • Isosceles: Two sides equal.
      • Scalene: All sides different.
  • Quadrilaterals:
    • Sum of angles = 360 degrees.
    • Types:
      • Square: All sides equal, all angles right.
      • Rectangle: Opposite sides equal, all angles right.
      • Parallelogram: Opposite sides and angles equal.
      • Trapezoid: One pair of parallel sides.
  • Circles:
    • Defined by radius, diameter, and circumference.
    • Area = πr², Circumference = 2πr.
• Theorems:
  • Pythagorean Theorem: In a right triangle, a² + b² = c², where c is the hypotenuse.
  • Congruence: Two shapes are congruent if they have the same size and shape.

Algebra

• Definition: Branch of mathematics dealing with symbols and rules for manipulating those symbols.
• Basic Concepts:
  • Variables: Symbols representing unknown values.
  • Expressions: Combination of variables, numbers, and operations.
  • Equations: Statements of equality.
  • Operations: Addition, subtraction, multiplication, and division are applied to algebraic expressions.
• Solving Equations:
  • Linear Equations: In the form ax + b = c.
  • Quadratic Equations: In the form ax² + bx + c = 0; solutions found using factoring, completing the square, or the quadratic formula.
• Functions:
  • Definition: A relation between a set of inputs and a set of outputs.
  • Types:
    • Linear Functions: f(x) = mx + b; graph is a straight line.
    • Quadratic Functions: f(x) = ax² + bx + c; graph is a parabola.
• Inequalities:
  • Represent a range of values rather than a single value.
  • Can be solved similarly to equations, but flipping the inequality sign is necessary when multiplying or dividing by a negative number.
• Polynomials:
  • Expressions involving variables raised to whole number powers.
  • Can be added, subtracted, multiplied, and divided.