Geometry and Measures covers shapes, angles, circle theorems, trigonometry (right-angled and non-right-angled), vectors, and mensuration — a significant portion of the exam.
Angles in parallel lines: alternate (Z), corresponding (F), co-interior (C, sum 180°). Polygon interior angle sum: (n−2) × 180°. Regular polygon exterior angle: 360°/n. Circle theorems: angle at centre = 2 × angle at circumference, angle in semicircle = 90°, opposite angles in cyclic quadrilateral = 180°, tangent perpendicular to radius, alternate segment theorem, angles in same segment equal.
Right-angled: SOH CAH TOA. sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj. Pythagoras: a²+b²=c². Exact values: sin30°=½, cos60°=½, sin45°=√2/2, tan45°=1. Non-right-angled: sine rule a/sinA = b/sinB, cosine rule a²=b²+c²−2bc cosA. Area = ½ab sinC. 3D trigonometry: identify right-angled triangles within 3D shapes.
Vectors: column notation, magnitude |v|=√(a²+b²), addition, scalar multiplication. Position vectors. Prove points are collinear: show one vector is scalar multiple of another. Area: rectangle, triangle (½bh), trapezium (½(a+b)h), circle (πr²). Volume: prism (area×length), cylinder (πr²h), cone (⅓πr²h), sphere (4/3πr³). Surface area. Similarity: linear scale factor k, area factor k², volume factor k³.
Sine rule: use when you have a matching pair (side and its opposite angle) plus one other piece. It solves: (1) two angles + one side (find other side), (2) two sides + non-included angle (find angle — beware ambiguous case). Cosine rule: use when you do NOT have a matching pair. It solves: (1) two sides + included angle (find third side), (2) all three sides (find an angle). Quick check: if you know the angle IN BETWEEN two known sides → cosine rule. If you have an angle NOT between the known sides → sine rule.
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