Plane and solid geometry
Plane and solid geometry questions ask you to combine standard theorems with a diagram you construct yourself. The exam favours triangle calculations, circle properties, similarity and congruence arguments, and coordinate methods for loci and distances.
Core rules
- Triangle: angles sum to 180°. Sine rule a/sin A=b/sin B=c/sin C. Cosine rule a^2=b^2+c^2-2bc cos A. Area = (1/2)bc sin A.
- Congruence needs SSS, SAS, ASA/AAS, or RHS. Similar triangles have equal corresponding angles and proportional sides; areas scale as the square of the ratio, volumes as the cube.
- Circle theorems: angle in a semicircle is 90°; angles in the same segment are equal; tangent is perpendicular to radius; intersecting chords give PA·PB=PC·PD.
- Inscribed solids: identify the relevant right triangle, often from centre to vertex and centre to face; use Pythagoras to find radii, heights, or slant heights.
- Coordinate geometry: distance, midpoint, section formula, gradients, and perpendicular gradients m1m2=-1.
Worked example
In triangle ABC, AB=5, AC=7, and angle A=60°. Find BC. Use the cosine rule with the side opposite angle A: BC^2=AB^2+AC^2-2(AB)(AC)cos A =25+49-2·5·7·(1/2)=74-35=39. Therefore BC=√39. A quick check: the side is shorter than 5+7 and longer than 7-5, and √39≈6.24 lies in that range.
Traps
The sine rule can produce an ambiguous SSA case; check whether a second triangle is possible. Do not use a similarity ratio directly for area or volume. In circle problems, identify which angle is subtended by which arc. For solids, mixing diameter and radius is a frequent error. When using coordinates, keep exact surds rather than premature decimals.