Module 3: Geometry, Trigonometry & Vectors
Master Circle Theorems, Trigonometry & Spatial Reasoning
Geometry accounts for 20% of Higher marks and 15% of Foundation marks. Spatial reasoning questions often require written geometric reasons — missing these statements will lose you communication marks!
1. Angle Rules & Polygon Geometry
- Angles on a straight line: Sum to 180°.
- Angles around a point: Sum to 360°.
- Vertically opposite angles: Are equal.
- Parallel Line Angles: Alternate angles are equal ("Z-angles"), Corresponding angles are equal ("F-angles"), Co-interior angles sum to 180° ("C-angles").
- Interior Angles of n-sided Polygon: Sum = (n - 2) × 180°.
- Exterior Angles of any Polygon: Always sum to 360°.
2. Essential Circle Theorems
| Circle Theorem Statement | Required Exam Reason Text |
|---|---|
| Angle at center vs circumference | "Angle at center is twice the angle at the circumference." |
| Angle in a semi-circle | "Angle subtended by a diameter at the circumference is 90°." |
| Angles in same segment | "Angles subtended by the same arc in the same segment are equal." |
| Cyclic Quadrilateral | "Opposite angles in a cyclic quadrilateral sum to 180°." |
| Tangent and Radius | "Tangent meets a radius at 90°." |
| Alternate Segment Theorem | "Angle between tangent and chord equals angle in alternate segment." |
In circle theorem exam questions, stating the exact theorem reason carries 50% of the total available marks for that question!
3. Pythagoras & Trigonometry Formulae
Right-Angled Triangles
- Pythagoras Theorem: a² + b² = c² (where c is the hypotenuse).
- SOH CAH TOA: sin θ = O/H, cos θ = A/H, tan θ = O/A.
Advanced Trigonometry (Higher Tier Non-Right Triangles)
| Formula Name | Equation | When to Use |
|---|---|---|
| Sine Rule | a / sin A = b / sin B = c / sin C | Given a matching side-angle pair. |
| Cosine Rule (Side) | a² = b² + c² - 2bc cos A | Given 2 sides and included angle (SAS). |
| Cosine Rule (Angle) | cos A = (b² + c² - a²) / (2bc) | Given all 3 sides (SSS). |
| Area of Triangle | Area = ½ a b sin C | Given 2 sides and included angle. |
4. Column Vectors & Vector Proofs (Higher)
A vector has both magnitude and direction. A column vector is written as (x, y)T.
- Vector Addition: (2, 3) + (-1, 4) = (1, 7).
- Vector Magnitude: |a| = √(x² + y²).
- Vector Proof of Parallel Lines: Prove vector AB = k × vector CD for a scalar multiplier k.
Frequently Asked Questions
Use the Cosine Rule when you know: (1) All three sides (SSS) and need an angle, or (2) Two sides and the included angle between them (SAS). Use the Sine Rule when you know a matching side-angle opposite pair!
The 3 most common are: (1) Angle at center is twice angle at circumference, (2) Angle in a semi-circle is 90 degrees, and (3) Opposite angles in a cyclic quadrilateral sum to 180 degrees. Always write the exact reason text for full marks!
To prove vectors AB and CD are parallel, show that vector CD is a scalar multiple of vector AB (i.e. CD = k × AB). If they also share a common point, they lie on a single straight line (collinear).