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IB Maths: Applications & Interpretation: Geometry and trigonometry

Voronoi diagrams, bearings, 3D measurement, matrices (HL). 16 subtopics, 64 key terms.

In this topic: 3.1 · 3.2 · 3.3 · 3.4 · 3.5 · 3.6 · 3.7 · 3.8 · 3.9 · 3.10 · 3.11 · 3.12 · 3.13 · 3.14 · 3.15 · 3.16

3.13D geometry: distance, volume and surface area

Key terms

Distance in 3D

d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)

Midpoint

Average the coordinates.

Right pyramid

Apex directly above the centre of the base.

Angle between a line and a plane

The angle between the line and its projection (shadow) on the plane.

Prisms and cylinders: \(V=\text{area of cross-section}\times\text{length}\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: The formula booklet has the volume and surface area formulas: practise finding them quickly rather than memorising.

3.2Trigonometry: right-angled and non-right-angled triangles

Key terms

SOH CAH TOA

Ratios for right-angled triangles.

Sine rule

a/sin A = b/sin B = c/sin C

Cosine rule

c² = a² + b² − 2ab cos C

Area

½ab sin C, with C the angle between sides a and b.

Use the sine rule when you know a side and its opposite angle.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Label the triangle in your working so the examiner can follow which side is a, b and c.

3.3Applications: elevation, depression and bearings

Key terms

Angle of elevation

Measured upwards from the horizontal.

Angle of depression

Measured downwards from the horizontal.

Bearing

Angle measured clockwise from north, written with three figures, e.g. 060°.

Back bearing

The bearing for the return journey: add or subtract 180°.

Always draw a labelled diagram first: mark north lines at each point, the known lengths and angles.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Give bearings with three figures (092°, not 92°).

3.4Arcs and sectors (degrees)

Key terms

Arc

Part of the circumference.

Sector

A “pizza slice” between two radii and an arc.

Arc length

(θ/360) × 2πr

Sector area

(θ/360) × πr²

The fraction \(\frac{\theta}{360}\) is the fraction of the full circle.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Read whether the question wants the arc length or the perimeter; the perimeter includes the two radii.

3.5Perpendicular bisectors

Key terms

Perpendicular bisector of AB

The line through the midpoint of AB at right angles to AB.

Equidistant

Every point on the perpendicular bisector is the same distance from A and from B.

Step 1: midpoint \(M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Check your answer: the midpoint must satisfy your equation.

3.6Voronoi diagrams

Key terms

Site

One of the given points (e.g. a hospital, a weather station).

Cell

The region of points closer to one site than to any other.

Edge

A part of a perpendicular bisector between two sites.

Vertex

A point where three or more edges meet: equidistant from three sites.

Nearest neighbour interpolation

Estimating a value at a point using the value of the site whose cell contains it.

Each cell contains exactly one site, and every point in a cell is nearest to that site.

The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.

Exam tip: In toxic-waste questions, list every candidate vertex with its distance to the nearest site, then choose the largest.

3.7Radians, arcs and sectors HL

Key terms

Radian

The angle where the arc length equals the radius.

Conversion

π radians = 180°

Arc length

l = rθ (θ in radians)

Sector area

A = ½r²θ (θ in radians)

Degrees to radians: multiply by \(\frac{\pi}{180}\). Radians to degrees: multiply by \(\frac{180}{\pi}\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: If an angle has no degree symbol, assume radians.

3.8Unit circle, identities and trigonometric equations HL

Key terms

Unit circle

Circle of radius 1: the point at angle θ is (cos θ, sin θ).

Pythagorean identity

cos²θ + sin²θ = 1

tan θ

sin θ ÷ cos θ

Signs by quadrant: all positive (1st), sin (2nd), tan (3rd), cos (4th).

The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.

Exam tip: Count the solutions on your GDC graph before writing your answer: missing one is the most common error.

3.9Matrix transformations HL

Key terms

Transformation matrix

A 2 × 2 matrix that maps (x, y) to a new point.

Composite transformation

Doing A then B is the single matrix BA.

Area scale factor

|det| of the transformation matrix.

Rotations are anticlockwise about the origin for positive \(\theta\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: For “A followed by B”, write BA: the first transformation sits next to the point.

3.10Vectors: concepts and notation HL

Key terms

Vector

Has magnitude and direction.

Position vector

The vector from the origin to a point.

Displacement vector AB

b − a

Unit vector

Magnitude 1: v ÷ |v|.

i, j, k

Unit vectors along the x, y, z axes.

Column form \(\begin{pmatrix}3\\-1\\2\end{pmatrix}\) is the same as \(3\mathbf i-\mathbf j+2\mathbf k\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Underline vectors in handwriting (or use arrows). Examiners need to see which quantities are vectors.

3.11Vector equation of a line and constant-velocity motion HL

Key terms

Vector equation of a line

r = a + λb: a point on the line plus multiples of a direction vector.

Direction vector

Gives the direction of the line.

Constant velocity

Position r = r₀ + tv; speed = |v|.

To check if a point is on a line, find \(\lambda\) from one component and test the others.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Speed is a number (the magnitude); velocity is a vector. Answer with the one the question asks for.

3.12Vectors in kinematics: variable velocity HL

Key terms

Displacement s(t)

Position vector as a function of time.

Velocity

v = ds/dt

Acceleration

a = dv/dt

Projectile

Motion with constant acceleration (0, −g).

Differentiate or integrate each component separately.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Remember the +C in each component, then use the initial position to find it.

3.13Scalar and vector products HL

Key terms

Scalar (dot) product

v · w = v₁w₁ + v₂w₂ + v₃w₃ = |v||w| cos θ: a number.

Perpendicular vectors

v · w = 0

Vector (cross) product

v × w: a vector perpendicular to both.

|v × w|

|v||w| sin θ = area of the parallelogram.

Area of a parallelogram \(=|\mathbf v\times\mathbf w|\); area of a triangle \(=\tfrac12|\mathbf v\times\mathbf w|\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: The dot product gives a number; the cross product gives a vector. Mixing them up costs all the marks.

3.14Graph theory: definitions HL

Key terms

Graph

A set of vertices (nodes) joined by edges.

Degree

The number of edges at a vertex (a loop counts 2).

Simple graph

No loops and no multiple edges.

Complete graph Kₙ

Every pair of the n vertices is joined: n(n − 1)/2 edges.

Tree

A connected graph with no cycles: n vertices, n − 1 edges.

Weighted / directed graph

Edges have values (distances, costs) / directions.

Handshaking lemma: the sum of the degrees \(=2\times\) the number of edges. So the number of odd-degree vertices is always even.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Use the handshaking lemma to check any graph you draw: an odd total of degrees means you made a mistake.

3.15Adjacency matrices and walks HL

Key terms

Adjacency matrix

Entry (i, j) = number of edges from vertex i to vertex j.

Walks of length k

The (i, j) entry of Aᵏ counts the walks of length k from i to j.

Weighted adjacency table

Entries are the weights of the edges.

Transition matrix (graph)

Directed graph where each column gives the probabilities of moving from that vertex.

An undirected graph has a symmetric adjacency matrix. Loops appear on the leading diagonal: follow the convention given in the question.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: State which power of A you used, then read the entry: (row = start, column = end).

3.16Graph algorithms: trees and routes HL

Key terms

Minimum spanning tree

Connects every vertex with the smallest total weight.

Kruskal's algorithm

Add the cheapest edge that doesn't make a cycle, repeat.

Prim's algorithm

Grow the tree from one vertex, always adding the cheapest edge to a new vertex (works on tables).

Chinese postman

Shortest closed route using every edge at least once.

Travelling salesman

Shortest closed route visiting every vertex: find upper and lower bounds.

Eulerian circuit (every edge once, back to start): exists if the graph is connected and every degree is even. Eulerian trail: exactly two odd vertices (start at one, finish at the other).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: In Kruskal and Prim questions, list the edges in the order you add them. That order is where the marks are.

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