Geometry and Trig Guide

Geometry: Area, Surface Area, and Volume

• 2D Shapes
      - Area: A measure of how many squares will fit into a shape, measured in units squared (${units^2}$).
      - Square: $Area = a^2$ or $Area = w \times h$.
      - Rectangle: $Area = w \times h$.
      - Triangle: $Area = \frac{b \times h}{2}$ or $Area = b \times h \times \frac{1}{2}$.
      - Circle: $Area = \pi r^2$.
      - Ellipse: $Area = \pi a b$.
      - Regular Polygon: $Area = \frac{n \times s \times a}{2}$, where $n$ is the number of sides.

• 3D Shapes
      - Surface Area (SA): A measure of the area of the outward facing sides.
      - Volume (V): A measure of how many cubes will fit in a shape, measured in units cubed (${units^3}$).
      - Cube: $SA = 6a^2$; $V = a^3$.
      - Sphere: $SA = 4 \pi r^2$; $V = \frac{4 \pi r^3}{3}$.
      - Cylinder: $SA = 2 \pi r h + 2 \pi r^2$; $V = \pi r^2 (h)$.
      - Pyramid: $SA = B + la$ and $V = \frac{Bh}{3}$. (Key: $B = \text{Base area}$, $la = \text{lateral area}$).
      - Prism: $SA = 2B + la$; $V = Bh$.

Geometry Vocabulary: Parallelograms

• Parallelogram: A quadrilateral in which BOTH pairs of opposite sides are parallel.

• Properties of Parallelograms:
      1. Opposite sides are congruent.
      2. Opposite angles are congruent.
      3. Consecutive angles ("next to" each other) are supplementary ($180^\circ$).
      4. Diagonals bisect each other.

• Proving Parallelograms in the Coordinate Plane:
      - Method 1: Prove both pairs of opposite sides are congruent ($=$). If $AB = DC$ and $AD = BC$, then $ABCD$ is a parallelogram.
          - Formula: Distance Formula $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
      - Method 2: Prove both pairs of opposite sides are parallel ($//$). If $AB // DC$ and $AD // BC$, then $ABCD$ is a parallelogram.
          - Formula: Slope Formula $m = \frac{y_2-y_1}{x_2-x_1}$.
      - Method 3: Prove one pair of opposite sides are congruent ($=$) and parallel ($//$). If $AB // DC$ and $AB = DC$, then $ABCD$ is a parallelogram.

Trigonometry Principles

• Definition: The study of relationships between the angles and the lengths of the sides of triangles.
• Notation: Angles are usually indicated with Greek letters: $\beta$ (beta), $\alpha$ (alpha), $\phi$ (phi), and $\theta$ (theta).
• Pythagoras Theorem: Used for calculating side lengths in right-angled triangles.
      - $c^2 = a^2 + b^2$
      - $a^2 = c^2 - b^2$
      - $b^2 = c^2 - a^2$
• Labeling Triangle Sides:
      - Hypotenuse: The longest side, located between the adjacent and opposite sides.
      - Opposite: The side opposite the main angle.
      - Adjacent: The side next to the angle, left over from the opposite and hypotenuse sides.
• Trigonometry Ratios: Applied to a right-angled triangle to define the relationship between sides and angles.
      - $\sin(\theta) = \frac{\text{opp}}{\text{hyp}}$
      - $\cos(\theta) = \frac{\text{adj}}{\text{hyp}}$
      - $\tan(\theta) = \frac{\text{opp}}{\text{adj}}$

Algebra: Probability, Combinations, and Permutations

• Combination: Use when the order of selection does not matter.
      - Formula: $C(n,r) = \frac{n!}{(n-r)! r!}$
      - Example: How many combinations for a dodgeball team can be formed from 24 students? Using $n=24$ and $r=6$: $C(24,6) = \frac{24!}{(24-6)! 6!} = \frac{24!}{18! 6!} = 134,596$ possible outcomes.
• Permutation: Use when the order of selection does matter.
      - Formula: $P(n,r) = \frac{n!}{(n-r)!}$
      - Example: Find the number of possible 4-digit locker combinations using digits 0-9. Using $n=10$ and $r=4$: $P(10,4) = \frac{10!}{(10-4)!} = \frac{3,628,800}{720} = 5,040$ possible outcomes.
• Probability: A ratio that compares how many times an outcome can occur with all possible outcomes.
      - $\text{Probability} = \frac{\text{desired outcomes}}{\text{possible outcomes}}$
• Types of Events:
      - Independent: Outcome of the first event does not influence the second. $P(X \text{ and } Y) = P(X) \times P(Y)$.
      - Dependent: Outcome of the first event influences the second. $P(X \text{ and } Y) = P(X) \times P(X \text{ after } Y)$.
      - Mutually Exclusive: Events cannot occur at the same time. $P(X \text{ or } Y) = P(X) + P(Y)$.
      - Not Mutually Exclusive: Events can happen at the same time. $P(X \text{ or } Y) = P(X) + P(Y) - P(X \text{ and } Y)$.
• Probability Example: If $P(\text{pink}) = \frac{1}{6}$, $P(\text{blue}) = \frac{1}{2}$, and $P(\text{green}) = \frac{1}{3}$, then $P(\text{pink or green}) = \frac{1}{6} + \frac{1}{3} = \frac{3}{6} = \frac{1}{2}$.

Advanced Algebraic Fractions and Polynomials

• Multiplication and Division of Fractions:
      1. Check for common factors.
      2. Factorize values.
      3. Cross cancel (only if there is one term on top and one term on bottom).
      - Example (from notes): $\frac{9-4x^2}{2x+3} \div \frac{2x^2-x-6}{x^3-8} \times \frac{x^2+3x+2}{x^2+2x+4}$ results in $\frac{-(x-2)(x+2)}{x+3}$.
• Addition and Subtraction of Fractions:
      1. Factorize denominators.
      2. Find the Lowest Common Denominator (LCD).
      3. Multiply numerators by the LCD.
      - The Butterfly Method: A visual shortcut for adding/subtracting fractions with unlike denominators. Multiply diagonal numerators and denominators ($3 \times 4 = 12$ and $1 \times 5 = 5$) and multiply denominators ($5 \times 4 = 20$) to get $\frac{17}{20}$.
• Polynomial Definitions:
      - Polynomial: An algebraic expression where variable exponents are non-negative integers.
      - Terms: Parts separated by $+$ or $-$ operators.
      - Monomial: Single term.
      - Binomial: Two terms.
      - Trinomial: Three terms.
      - Like Terms: Have the same variable and power.
      - Standard Form: Written with exponents in descending order (e.g., $3x^4 + 2x^3 + 8x^2 + 4x + 1$).
• Polynomial Operations:
      - Adding: Add like terms.
      - Subtracting: Rewrite as addition using "Keep Change Change" ($KCC$). Change the sign of every term in the subtracted polynomial.
      - Multiplying (FOIL): First, Outside, Inside, Last. Example: $(5x+3)(2x+1) = 10x^2 + 5x + 6x + 3 = 10x^2 + 11x + 3$.

General Geometry and Angles

• Basic Terminology:
      - Polygon: A closed shape with no curved lines.
      - Vertex: Where two sides meet.
      - Face: Flat surface of a 3D shape.
      - Edge: Line segment where two faces meet.
      - Linear Pairs: Form a straight line and are supplementary ($180^\circ$).
      - Complementary Angles: Sum to $90^\circ$.
      - Supplementary Angles: Sum to $180^\circ$.
• Angles from Parallel Lines cut by a Transversal:
      - Vertical Angles: Formed at intersections, always equal.
      - Corresponding Angles: Coincide if line positions are overlaid, always equal.
      - Alternate Interior/Exterior: On opposite sides of the transversal, always equal.
      - Consecutive Interior: On the same side of the transversal, supplementary ($180^\circ$).
• Coordinate Graphs:
      - Midpoint Formula: $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$.
      - Gradient (Slope): $m = \frac{y_2-y_1}{x_2-x_1}$. Parallel lines have the same gradient; perpendicular lines have the negative reciprocal gradient.
      - Length (Distance): Uses the formula $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
      - Example with coordinates (4,4) and (12,2):
          - Midpoint: $(8,3)$.
          - Length: $\approx 8.37\,\text{cm}$.
          - Gradient: $-1/4$.
          - Equation: $y = -1/4x + 5$.
          - Perpendicular through (2,1): $y = 4x - 9$.

Circle Geometry and Sector Formulas

• Circle Components:
      - Arc: A portion of the circumference.
      - Chord: A line segment joining two points on the circle.
      - Sector: A "slice" of the circle bounded by two radii and an arc.
• Arc Length Formulas:
      - Whole Circle: $C = \pi d$.
      - Arc Length: $\frac{\text{angle}}{360} \times \pi d$.
      - Example ($30^\circ$, $r=15$): $Arc = \frac{30}{360} \times \pi \times 30 = 7.85\,\text{cm}$.
• Sector Area Formulas:
      - Generic Formula: $Sector\,Area = \frac{\text{angle}}{360} \times \pi r^2$.
      - Quarter Circle: $Area = \frac{1}{4} \pi r^2$.
      - Example ($36^\circ$, $r=20$): $Area = \frac{36}{360} \times \pi \times 400 = 125.7\,\text{cm}^2$.
      - Example ($270^\circ$, $r=5$): $Area = \frac{270}{360} \times \pi \times 25 = 58.9\,\text{m}^2$.

Circle Theorems

• Rule 1: Angles in the same segment (subtended by the same chord) are equal.
• Rule 2: Opposite angles in a cyclic quadrilateral sum to $180^\circ$.
• Rule 3: The angle at the center is twice the angle at the circumference ($x$ vs $2x$).
• Rule 4: The perpendicular bisector of a chord passes through the center of the circle.
• Rule 5: The radius meets the tangent at $90^\circ$.
• Rule 6: Tangents from the same exterior point to a circle are equal in length.
• Rule 7: The angle inscribed in a semicircle is always a right angle ($90^\circ$).
• Rule 8 (Alternate Segment Theorem): The angle between a tangent and a chord is equal to the angle in the alternate segment.
• Generic Summary Points:
      - Center Properties: Three properties (through center, bisect chord, perpendicular to chord) — any two prove the third.
      - Tan-Radius Theorem: Angle between radius and tangent is $90^\circ$.
      - Tan-Chord Theorem: Also known as the Alternate Segment Theorem.

Standard Mathematics Formulas

• Trigonometry Rules:
      - Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.
      - Cosine Rule: $a^2 = b^2 + c^2 - 2bc \cos(A)$ or $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$.
      - Area of a Triangle: $Area = \frac{1}{2} ab \sin(c)$.
• Trig Values:
      - $\sin(30^\circ) = 1/2$
      - $\sin(45^\circ) = 1/\sqrt{2}$
      - $\cos(60^\circ) = 1/2$
      - $\tan(45^\circ) = 1$
• Financial Math:
      - Simple Interest: $I = P \times r \times t$.
      - Compound Interest: $A = P(1 + \frac{r}{100})^{\text{years}}$.
• Algebraic Formulas:
      - Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
      - Trapezium Rule: $Area = \frac{1}{2}(a+b)h$.