Geometry & Trigonometry: notes and practice questions

2D Distance between points $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$. 2D Midpoint of points $(x_1, y_1)$ and $(x_2, y_2)$: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.

Volume: Measure of 3D space a shape occupies. Surface Area: Total area of all outer faces and curved surfaces of a 3D object.

Hypotenuse (H): Longest side, opposite $90^\circ$. Opposite (O): Side directly across from reference angle $\theta$.

Label triangles with capital letters for angles (A, B, C) and corresponding lowercase letters for opposite sides (a, b, c). Relabel triangle vertices as needed, maintaining consistent opposite side-angle pairs.

For right-angled triangles: Pythagoras' Theorem: $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse).

Arc: Part of a circle's circumference. Minor arc ($\theta < 180^\circ$ or $\pi$ rad), Major arc ($\theta > 180^\circ$ or $\pi$ rad). Sector: Region of a circle enclosed by two radii and an arc. Minor sector ($\theta < 180^\circ$), Major sector ($\theta > 180^\circ$).

A perpendicular bisector cuts a line segment in half at $90^\circ$, with every point on it equidistant from the segment's endpoints. Perpendicular Gradients:.

A Voronoi diagram partitions a 2D plane into regions (cells) based on the distance to a set of "sites" (specific points). Each Voronoi cell contains all points closer to its site than to any other site.

Radian: Angle subtended at circle's center by an arc equal to the radius. Fundamental relationship: $2\pi \text{ rad} = 360^\circ$ or $\pi \text{ rad} = 180^\circ$.

Unit Circle: radius 1, centered at $(0,0)$. Angle Measurement: positive angles are anti-clockwise from positive x-axis.

Position vector for a point: $\begin{pmatrix} x \\ y \end{pmatrix}$. Position matrix for multiple points: $P = \begin{pmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{pmatrix}$.

Scalar: quantity with magnitude only. Vector: quantity with magnitude and direction. Vector notation: bold lowercase (e.g., a) or arrow over points (e.g., $\vec{AB}$).

A vector equation of a line defines a path using a starting position and a direction. Position Vector: Describes a point's location relative to the origin.

Kinematics: branch of mathematics modeling and analyzing object motion. Scalars: quantities with magnitude only (e.g., time, speed, distance).

Scalar product (dot product) of two vectors yields a scalar (real number). Denoted as $v \cdot w$.

Vector Equation of a Line: $r = a + \lambda b$. $r$: position vector of any point on the line.

Vertex (Node): A point representing an object or place. Edge (Arc): A line connecting two vertices.

Adjacency Matrix: Represents graph connections between vertices. Complete Graph: Each vertex connected to every other vertex by a single edge.

Graph Terminology:. Complete Graph: Each vertex connected to every other.