Geometry Basics Quiz

Geometry

• Definition: Study of shapes, sizes, and properties of space.
• Key Concepts:
  • Points: No dimension; location in space.
  • Lines: Infinite length; no width; determined by 2 points.
  • Angles: Formed by two rays (sides) with a common endpoint (vertex).
    • Types: Acute (< 90°), Right (90°), Obtuse (> 90°), Straight (180°).
  • Triangles: Three-sided figures.
    • Types: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal).
    • Sum of angles = 180°.
  • Quadrilaterals: Four-sided figures.
    • Types: Square, Rectangle, Parallelogram, Trapezoid.
    • Sum of angles = 360°.
  • Circles: Set of points equidistant from a center.
    • Key terms: Radius, Diameter, Circumference, Area (A = πr²).
  • Polygons: Closed figures with straight sides.
    • Regular polygons: All sides and angles equal.
• Theorems:
  • Pythagorean theorem: In right triangles, (a^2 + b^2 = c^2).

Algebra

• Definition: Branch of mathematics dealing with symbols and rules for manipulating those symbols.
• Key Concepts:
  • Variables: Symbols representing unknown values (e.g., x, y).
  • Expressions: Combinations of numbers, variables, and operators (e.g., 2x + 3).
  • Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
  • Functions: Relations where each input has exactly one output (e.g., f(x) = x²).
  • Linear equations: Equations of the first degree (e.g., y = mx + b).
• Operations:
  • Addition, Subtraction: Combining like terms.
  • Multiplication, Division: Distributing, factoring.
  • Solving equations: Finding the value of variables.
• Key Formulas:
  • Slope-intercept form: (y = mx + b) (m = slope, b = y-intercept).
  • Quadratic formula: For (ax² + bx + c = 0), roots are (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
• Inequalities: Expressions showing the relationship of values (e.g., x > 5).
• Polynomials: Expressions with multiple terms.
  • Types: Monomial, Binomial, Trinomial.
  • Operations include addition, subtraction, multiplication, and factoring.

Geometry

• Points are dimensionless locations in space
• Lines have infinite length, no width, and are defined by two points
• Angles are formed by two rays with a common endpoint, called the vertex.
  • Acute angles: less than 90 degrees.
  • Right angles: exactly 90 degrees.
  • Obtuse angles: greater than 90 degrees.
  • Straight angles: 180 degrees.
• Triangles are three-sided figures.
  • Equilateral triangles: all sides equal.
  • Isosceles triangles: two sides equal.
  • Scalene triangles: no sides equal.
  • The sum of all interior angles in a triangle is always 180 degrees.
• Quadrilaterals are four-sided figures with a total of 360 degrees.
  • Squares: all sides equal, all angles are right angles.
  • Rectangles: opposite sides equal, all angles are right angles.
  • Parallelograms: opposite sides parallel, opposite angles equal.
  • Trapezoids: one pair of opposite sides parallel.
• Circles are sets of points equidistant from a center point.
  • Radius: distance from the center to any point on the circle.
  • Diameter: distance across the circle through the center (twice the radius).
  • Circumference: the distance around the circle.
  • Area of a circle: πr² (pi times the radius squared)
• Polygons are closed figures with straight sides.
  • Regular polygons: all sides and angles equal.
• Pythagorean Theorem: in right triangles, (a^2 + b^2 = c^2) where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse (longest side).

Algebra

• Variables are letters that represent unknown values
• Expressions are combinations of variables, numbers, and mathematical operations like addition, subtraction, multiplication, and division
• Equations state that two expressions are equal to each other.
• Functions are relationships where each input has exactly one output.
• Linear Equations are equations of the first degree, often written as y = mx + b.
  • Slope (m) represents how steeply the line rises or falls.
  • Y-intercept (b) is the point where the line crosses the y-axis.
• Solving Equations involves finding the value of the variable that makes the equation true.
• Quadratic Formula: Used to solve quadratic equations (equations involving x²).
  • For (ax² + bx + c = 0), the roots (solutions) are (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
• Inequalities show the relationship between values using symbols like > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to)
• Polynomials: Expressions with multiple terms in one variable.
  • Monomial: one term (e.g., 2x)
  • Binomial: two terms (e.g., x² + 2x)
  • Trinomial: three terms (e.g., 3x² + 2x + 1)
  • Addition, subtraction, multiplication, and factoring are common operations with polynomials.