Unit 4 extends geometry to situations where:
- Triangles are not right-angled (use Sine Rule or Cosine Rule)
- Problems require multiple steps combining similarity and trigonometry
- Three-dimensional figures require 2D cross-section analysis
For any triangle with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Use when you know: two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA — beware the ambiguous case).
In triangle PQR: $PQ = 12$ cm, $\angle P = 55°$, $\angle Q = 70°$. Find $QR$.
$\angle R = 180° - 55° - 70° = 55°$.
$$\frac{QR}{\sin P} = \frac{PQ}{\sin R} \implies QR = \frac{12 \sin 55°}{\sin 55°} = 12 \text{ cm}$$
(Triangle PQR is isosceles since $\angle P = \angle R$.)
For any triangle:
$$c^2 = a^2 + b^2 - 2ab\cos C$$
$$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$
Use when you know: three sides (SSS), or two sides and the included angle (SAS).
A triangular plot of land has sides 80 m and 65 m with an included angle of 42°. Find the third side.
$$c^2 = 80^2 + 65^2 - 2(80)(65)\cos(42°)$$
$$= 6400 + 4225 - 10400 \times 0.7431 = 10625 - 7728 = 2897$$
$$c = \sqrt{2897} \approx 53.8 \text{ m}$$
When scale factor $k$ is applied, areas scale by $k^2$ and volumes by $k^3$.
Two similar cones have base radii 4 cm and 10 cm. The smaller cone has volume \$67.0 \text{ cm}^3$. Find the volume of the larger.
$$k = \frac{10}{4} = 2.5 \implies \text{Volume ratio} = 2.5^3 = 15.625$$
$$V_{\text{large}} = 67.0 \times 15.625 \approx 1047 \text{ cm}^3$$
Many 3D problems (e.g. angles of elevation in buildings, slope angles, diagonal measurements) require:
1. Identify a right-angled triangle in a 2D cross-section
2. Find a key length using Pythagoras or basic trig
3. Use that length in a second triangle (possibly non-right-angled)
COMMON MISTAKE: Applying basic SOH-CAH-TOA to a non-right-angled triangle. Always check for a right angle before using basic trig ratios. If no right angle, use Sine Rule or Cosine Rule.
EXAM TIP: For non-right-angled triangles, decide: known two angles + one side → Sine Rule; known two sides + included angle (or all three sides) → Cosine Rule.
