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.