Geometry and Trigonometry — IB Math AI HL Study Guide

For: IB Mathematics Applications & Interpretation HL candidates.

Covers: IB AI HL Topic 3 — coordinate geometry, sine/cosine rules, area of triangle, 3D distances and angles, voronoi diagrams, vectors, graph theory (HL extension), applications-focused examples.

You should already know: Basic trigonometry, Pythagoras' theorem.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB AI HL style for educational use. They are not reproductions of past IBO papers.

1. Why Geometry & Trig in AI HL

Topic 3 is one of 5 syllabus topics in AI HL. The "AI" in the syllabus name signals that everything is applications-driven — geometry problems involve real maps, surveying, navigation, GPS distances. About 18-22% of your IBO assessment touches Topic 3 (split across Paper 1, Paper 2, IA modelling).

HL extension topics not in SL: vectors, graph theory, adjacency matrices, transition matrices for graphs.

2. Trigonometry of triangles

For any triangle with sides $a,b,c$ opposite angles $A,B,C$:

Sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Use when you have AAS, ASA, or SSA configurations.

Cosine rule: $c^2=a^2+b^2-2ab\cos C$. Use when you have SAS or SSS.

Area: $\text{Area}=\frac{1}{2}ab\sin C$.

Ambiguous case (SSA): when given two sides and a non-included angle, there can be 0, 1, or 2 valid triangles. Always check by computing both possible angles.

3. 3D geometry

Distance between points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$: $$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$$

Angle between a line and a plane, or two planes, often requires constructing perpendicular projections. Sketch first, then apply trig in the resulting right triangle.

Volumes of standard solids are exam-given: cone $\frac{1}{3}\pi r^{2}h$, sphere $\frac{4}{3}\pi r^3$, etc. Surface areas similar.

4. Voronoi diagrams (AI specific)

A Voronoi diagram partitions a plane based on proximity to a set of "site" points. Each region contains all points closer to one site than to any other.

Construction:

1. For each pair of sites, draw the perpendicular bisector of the segment joining them.
2. Each cell is the intersection of half-planes — the region nearer to its site.

Common AI HL problems: assign customers to nearest store, locate a new store to maximise market share, or partition emergency-response zones.

5. Vectors (HL extension)

Position vector of point $P$: $\overrightarrow{OP}$.

Vector between two points: $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$.

Magnitude: $|\vec{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$.

Dot product: $\vec{a}\cdot \vec{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3=|\vec{a}||\vec{b}|\cos \theta$.

Use dot product to find the angle between vectors: $\cos \theta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}$.

Vector equation of line: $\vec{r}=\vec{a}+t\vec{d}$, where $\vec{a}$ is a point on the line and $\vec{d}$ is the direction vector.

6. Graph theory (HL extension)

A graph has vertices (nodes) and edges connecting them. Key terms:

• Degree of a vertex: number of edges connected to it.
• Path: sequence of edges connecting vertices.
• Cycle: path that starts and ends at the same vertex.
• Tree: connected graph with no cycles.
• Adjacency matrix: $A_{ij}=1$ if there's an edge from $i$ to $j$, 0 otherwise.

The matrix product $A_{ij}^n$ gives the number of paths of length exactly $n$ from $i$ to $j$. This is a frequent IB AI HL Paper 2 question.

Minimum spanning tree (Prim's or Kruskal's algorithm): find a tree connecting all vertices with minimum total edge weight. Used for laying cable, routing roads.

7. Worked Example

Three radio masts $A$, $B$, $C$ have positions $A(0,0)$, $B(8,0)$, $C(4,6)$ on a flat map (km).

(a) Use the cosine rule to find angle $BAC$. (b) Use the sine rule to find the side $BC$ — verify with distance formula. (c) Find the position of the point equidistant from $A$ and $B$ on the segment closest to $C$.

Solution.

(a) $|AB|=8$, $|AC|=\sqrt{16+36}=\sqrt{52}=2\sqrt{13}\approx 7.21$, $|BC|=\sqrt{16+36}=7.21$. By cosine rule: $|BC{|}^2=|AB{|}^2+|AC{|}^2-2|AB||AC|\cos A$. $52=64+52-2(8)(7.21)\cos A$. $\cos A=(64+52-52)/(2\cdot 8\cdot 7.21)=64/115.4=0.554$, so $A\approx 56.3°$.

(b) Distance formula: $|BC|=\sqrt{(8-4{)}^2+(0-6{)}^2}=\sqrt{16+36}=7.21$ ✓.

(c) Perpendicular bisector of $AB$ is the line $x=4$. The point on $x=4$ closest to $C(4,6)$ is $C$ itself, but $C$ is not on segment $AB$. The point on segment $AB$ closest to $C$ is $(4,0)$ — drop perpendicular from $C$ to $AB$. So $(4,0)$ is equidistant from $A$ and $B$ (both 4 km away).

8. Common Pitfalls

• Sine rule ambiguous case: when using ${\sin }^{-1}$, the calculator returns only the acute angle. Check whether the obtuse possibility (180° − that) is also valid given the other constraints.
• Confusing degree and radian modes: IB papers can use either. Always check what your calculator is in.
• Voronoi regions are open at infinity: peripheral cells extend infinitely; only interior cells are bounded.
• Vector arithmetic: $\overrightarrow{AB}=B-A$ (head minus tail), not $A-B$.

9. Practice Questions

1. In triangle $XYZ$, $|XY|=12$, $|XZ|=9$, angle $X=50°$. Find $|YZ|$.
2. Three towns $P,Q,R$ at $(0,0),(5,0),(3,4)$. Construct the Voronoi cell containing $P$ and describe it.
3. Vectors $\vec{a}=(3,4,0)$ and $\vec{b}=(1,2,2)$. Find $\vec{a}\cdot \vec{b}$ and the angle between them.

10. Quick Reference Cheatsheet

• Sine rule: $a/\sin A=b/\sin B=c/\sin C$.
• Cosine rule: $c^2=a^2+b^2-2ab\cos C$.
• Triangle area: $\frac{1}{2}ab\sin C$.
• 3D distance: $\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}$.
• Voronoi: perpendicular bisectors → nearest-site regions.
• Dot product: $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$.
• Adjacency matrix power $A^n$: count paths of length $n$.

11. What's Next

Geometry & Trig in AI HL is closely linked to Topic 4 (Statistics) — many real-world problems combine both (e.g. survey area + sample mean). Use Ollie for any specific applied problem: "Help me set up the cosine rule for this surveying question" or "Walk me through the Voronoi construction step by step."