Geometry and Trigonometry
AA HL Topic 3 (3.1, 3.2, 3.4, 3.5, 3.8)
Three tools run through almost every mark in this topic: the distance/midpoint/gradient toolkit for anything on coordinate axes, the unit-circle definitions and identities for anything involving an angle, and the sine/cosine rules for triangles that refuse to be right-angled. Examiners recycle the same traps every session — wrong perpendicular gradient, forgetting the second root of a modulus equation, picking the wrong form of , missing the ambiguous case. Learn to spot which trap a question is baiting before you touch your calculator.
Overview
This chapter has two personalities. Coordinate geometry lives on the -plane and turns every geometric claim (perpendicular, equal length, equal area, collinear) into an algebraic equation you solve. Trigonometry lives on the unit circle and turns every angle into a ratio you can manipulate with identities. The sine and cosine rules are the bridge: they let you attack a triangle with no right angle using exactly the same coordinate data (side lengths, angles) that coordinate geometry hands you.
- Coordinate geometry questions almost always chain three or four sub-parts: find a length/gradient, find a midpoint, then use both to prove something (perpendicular bisector, equal area, parallel median).
- Trig identity questions are proof questions in disguise — the command term is usually 'show that' or 'prove', and marks are for algebraic justification, not just a correct final line.
- Sine/cosine rule questions are almost always two-step: find one unknown with cosine rule (or sine rule), then feed it into the other rule for the next unknown.
The shape of the chapter
Command terms this topic tests hardest
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Find / Determine | Obtain the answer, showing enough working to justify it | AO2 | Method mark for correct formula substitution even with an arithmetic slip; final mark needs the exact simplified value. |
| Show that | Prove a given result via logical steps; the last line must state the printed answer exactly | AO2/AO3 | No credit for jumping straight to the given answer — every algebraic line needs to be visible. |
| Hence | Use the immediately preceding result or method directly | AO3 | An independent method that reaches the right answer without using the earlier result still loses the 'hence' mark. |
| Deduce | State a conclusion using previous reasoning, with minimal new calculation | AO3 | Full marks need an explicit reasoning link ('since the median bisects the area...'), not a full recomputation. |
| Prove | Construct a general argument valid for all cases, using variables not specific numbers | AO2/AO3 | Checking one numerical example instead of general algebra scores 0 out of the marks available. |
| Justify | Give a reason that supports a stated fact or choice | AO3 | A bare assertion with no link to a named theorem or identity loses the mark even when the conclusion is correct. |
Key point
Overview