IP AMaths Notes (Upper Sec, Year 3-4): 08) Plane Geometry with Algebra

Upper-sec paper setters mix Euclidean geometry with algebraic coordinates. Treat every diagram as an algebra system: assign variables, write equations, then solve.

Key reminders

• Interior angles of a triangle sum to \( \pi \) rad (\( 180^\circ \)).
• Circle theorems still apply with coordinates: equal chords subtend equal angles, angle at centre is twice the angle at circumference, cyclic quadrilaterals have opposite angles summing to \( \pi \).
• Similar triangles justify proportions such as \( \dfrac{AB}{DE} = \dfrac{AC}{DF} \) when \( \triangle ABC \sim \triangle DEF \).
• Right triangles unlock Pythagoras and the primary trig ratios.

Worked example 1 — Coordinate angle chase

In \( \triangle ABC \), points \( A(0, 0) \), \( B(6, 0) \), and \( C(2, 4) \). Find \( \angle ACB \) in radians.

1. Form vectors \( \vec{CA} = (-2, -4) \) and \( \vec{CB} = (4, -4) \).
2. Dot product: \( \vec{CA} \cdot \vec{CB} = (-2)(4) + (-4)(-4) = 8 \).
3. Magnitudes: \( \lVert \vec{CA} \rVert = \sqrt{(-2)^2 + (-4)^2} = \sqrt{20} \); \( \lVert \vec{CB} \rVert = \sqrt{4^2 + (-4)^2} = \sqrt{32} \).
4. Hence \( \cos \angle ACB = \dfrac{8}{\sqrt{20}\sqrt{32}} = \dfrac{1}{\sqrt{10}} \).
5. So \( \angle ACB = \arccos\bigl(10^{-1/2}\bigr) \approx 1.249 \) rad.

Worked example 2 — Similar triangles with algebra

An isosceles triangle has equal sides of length \( x \) and base \( 6 \). Its height is \( 4 \). Find \( x \) and the vertex angle.

1. Altitude splits the base: each half is \( 3 \).
2. Right triangle gives \( x^2 = 4^2 + 3^2 = 25 \) so \( x = 5 \).
3. Let vertex angle be \( \theta \); then \( \cos \dfrac{\theta}{2} = \dfrac{3}{5} \).
4. Therefore \( \theta = 2 \arccos \left(\dfrac{3}{5}\right) = 2.214 \) rad (3 s.f.).

Try this

Construct coordinates for four points on a circle of radius \( 5 \) whose central angles are \( 70^\circ \), \( 110^\circ \), \( 140^\circ \), and \( 40^\circ \). Verify numerically that opposite angles of the resulting cyclic quadrilateral sum to \( \pi \).