3.3 · Geometry and measures

OxfordAQA 9260 unit 3.3 (specification Version 5.1) — at 31.5% of the combined specimen marks. Core rows G1–G26 cover angle facts through constructions, mensuration, similarity, trigonometry, transformations, vectors and matrices; the Extension additions add the full circle-theorem programme with proof, congruence conditions, bearings, sphere/pyramid/cone/frustum formulae, k-k²-k³ similarity, sectors, 3D Pythagoras and trigonometry, the sine/cosine rules with ½ab sinC, combined transformations with fractional/negative scale factors, vector geometry, and the 2×2 matrix programme (identity, unit-square transformations, combinations). The 1E specimen is the geometry paper; unsampled lines are labelled specification originals.

Must-know for unit 3.3 (NO formulae sheet):

Conventional terms: points, lines, vertices, edges, planes; parallel, perpendicular; polygons 多边形 and regular polygons 正多边形; the standard triangle labelling. Angle facts: at a point (360°), on a line (180°), vertically opposite 对顶角; parallel lines give alternate 内错角, corresponding 同位角 and co-interior angles (use these names in reasons, never "Z angles"); triangle sum 180° with the isosceles 等腰, equilateral 等边 and scalene 不等边 families. Polygon sums: interior $(n-2) \times 180°$, exterior $360°$; for regular polygons also the angle at the centre $= \tfrac{360°} {n}$. The special quadrilaterals (square, rectangle, parallelogram, trapezium, kite, rhombus 菱形) carry their side/angle/diagonal properties, and 2D shapes have reflection and rotation symmetry 对称 counts (a regular hexagon: 6 lines, order 6).

Congruence 全等 with the Extension conditions for triangles — SSS, SAS, ASA, RHS — each justified by which three independent measurements pin a triangle. Similarity 相似 scales lengths by $k$: (Extension) areas by $k^2$, volumes by $k^3$, in every direction.

The Extension circle programme: angle at the centre is twice the angle at the circumference; the semicircle 半圆 angle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral 圆内接四边形 sum to 180°; tangent $\perp$ radius 半径; tangents from an external point are equal; the perpendicular from the centre bisects a chord 弦; the alternate segment 弦切角 theorem. The verified chain: an angle at the circumference of 35° subtends the same arc as a centre angle of 70°, and the angle from the OPPOSITE arc is $\tfrac12(360° - 70°) = 145°$ — the two circumference angles are opposite angles of a cyclic quadrilateral ($35° + 145° = 180°$) — every step names its theorem (the 1E specimen's circle question carries answers of this family; G9's geometric proof 证明 demand).

Faces, surfaces, edges and vertices of cubes, cuboids, prisms 棱柱, cylinders 圆柱, pyramids, cones and spheres; plans and elevations 平面图和立面图 (front, side, top views — a specification line not sampled by the specimens). Measure line segments and angles; scale drawings with scale factors; bearings 方位角: three-figure, clockwise from north (Extension includes the eight compass points) — the 2E specimen's 252° comes from a bearings chain: a return bearing plus an angle at a point.

Worked example: an angle from two bearings. $B$ is on a bearing of 070° from $A$, and $C$ is on a bearing of 160° from $B$. Find angle $ABC$. The bearing of $A$ FROM $B$ is the return bearing $070° + 180° = 250°$. At $B$, the direction to $A$ is 250° and the direction to $C$ is 160°, so angle $ABC = 250° - 160° = 90°$. Draw a north line at EVERY point you work from; bearings are measured at the point named after "from".

Ruler-and-compass constructions 作图: perpendicular bisector 垂直平分线 of a segment, perpendicular to a line from/at a point, angle bisector 角平分线, 60° angle — used to build figures and solve loci 轨迹 problems; the perpendicular distance from a point to a line is the shortest. Leave all arcs visible: the arcs are the working.

Standard units with conversions (time 12/24-hour; length, area, volume/capacity, mass; compound units 复合单位 speed and density 密度). Areas: triangle $\tfrac12 bh$, parallelogram $bh$, trapezium $\tfrac12(a+b)h$; volume by $V = Ah$ (constant cross-section 横截面). The circle: $2\pi r$, $\pi r^2$; (Extension) sectors 扇形 arc $\tfrac\theta{360} \times 2\pi r$ and area $\tfrac\theta{360} \times \pi r^2$, plus sphere/pyramid/cone surface and volume formulae and composite solids including frustums 圆台 (specification lines, several not sampled) — a cone of $r = 5$, $h = 12$ has slant 13, $V = 100\pi$ and curved surface $65\pi$; the 1E specimen's cone problem finds a proportion $p = 15$.

Worked example: a sector. A sector has radius 6 cm and angle 150°. The arc is $\tfrac{150}{360} \times 2\pi \times 6 = 5\pi = 15.7$ cm, so the perimeter is $5\pi + 6 + 6 = 27.7$ cm (both radii count). The area is $\tfrac{150}{360} \times \pi \times 6^2 = 15\pi = 47.1\ \text{cm}^2$. Similar figures (Extension): lengths $k$ : areas $k^2$ : volumes $k^3$ — the 2E specimen's similar solids give 864 cm³ from a volume ratio.

$a^2 + b^2 = c^2$ with SOH CAH TOA in right-angled triangles — 2D always, 3D figures at Extension: extract the right triangle first (the 1E specimen's 3D trig gives 62.7°). The sine rule 正弦定理 $\dfrac a{\sin A} = \dfrac b{\sin B}$ and cosine rule 余弦定理 $a^2 = b^2 + c^2 - 2bc\cos A$ cover any triangle (Extension), with Area $= \tfrac12 ab\sin C$ reversible — the 2E specimen's cosine rule gives 102.6°, and $\cos x = \tfrac8{11}$ yields $x = 43.3°$.

Worked example: 2E cosine-rule family, adapted. A triangle has $a = 9$, $b = 11$ and angle $C = 72°$. $c^2 = 81 + 121 - 2(9)(11)\cos 72° = 202 - 198 \times 0.309 = 202 - 61.2 = 140.8$, so $c = 11.9$ (3 s.f.). Then the smallest angle sits opposite the shortest side: $\cos A = \tfrac{b^2 + c^2 - a^2}{2bc} = \tfrac{121 + 140.8 - 81}{2 \times 11 \times 11.87} = \tfrac{180.8}{261.1} = 0.693$, so $A = 46.2°$ (keeping $c$ unrounded) — check $A + B + C = 180°$ with $B = 61.8°$. ✓

Single rotations, reflections, translations (by a vector) and enlargements (positive factor) — with the preserved properties named; the Extension additions cover combined transformations and enlargements by fractional and negative scale factors 负比例因子 (an inverted, shrunken image on the opposite side of the centre). Vectors 向量: notation, sum, difference and scalar multiples (graphically and by components), the commutative and associative laws, and simple 2D geometry — the 2E specimen proves $CB \parallel MN$ via a $\tfrac12$ ratio using vectors.

The 2×2 matrix 矩阵 programme (Extension): multiplication (2×2 by 2×2, by 2×1, scalar), the identity matrix 单位矩阵 $I$; transformations of the UNIT SQUARE — rotations of 90°/180°/270° about the origin, reflections in $x = 0$, $y = 0$, $y = x$, $y = -x$, enlargements centred on the origin — represented by matrices such as $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ (reflection in $y = -x$, the 1E specimen's matrix); combined transformations by matrix multiplication, in the order the transformations apply.

Worked example: a combined transformation. Reflect in $y = x$, matrix $\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, then rotate 90° anticlockwise about the origin, matrix $\mathbf{R} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. The FIRST transformation sits on the RIGHT:

which is a reflection in the $y$-axis ($x = 0$). Check with a point: $(2, 1)$ reflects to $(1, 2)$, which rotates to $(-2, 1)$, and the reflection in the $y$-axis also sends $(2, 1)$ to $(-2, 1)$. ✓ Multiplying in the other order gives $\mathbf{M}\mathbf{R}$, a reflection in the $x$-axis: the order matters.

Worked example: 1E matrix-solve family, adapted. Solve $\begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 10 \\ 5 \end{pmatrix}$: the equations are $3x + y = 10$ and $x + 2y = 5$. From the first, $y = 10 - 3x$; substituting: $x + 2(10 - 3x) = 5$, so $-5x = -15$, giving $x = 3$ and $y = 1$. Check: $3(3) + 1 = 10$. ✓ The 1E specimen's own pairs $(6, -\tfrac12)$ and $(-2, -\tfrac52)$ come from the same method on a second, linked system.

• Conventional terms and triangle notation; angle facts; parallel-line angles; polygon interior/exterior/centre sums; quadrilateral properties; symmetry counts (G1–G6).
• Congruence with SSS/SAS/ASA/RHS; similarity with k, k², k³ ratios (G7, G17, Extension).
• Circle language and the full theorem bank with proof-style reasons (G8–G9, Extension; 35° → 70° → 145° verified chain).
• 3D shape properties; plans and elevations (not sampled); measuring, scale factors, bearings (G10–G12).
• Constructions (four ruler-and-compass types) with loci; shortest distance property (G13).
• Units and compound measures; area and V = Ah formulae; circle formulae; sectors; sphere/pyramid/cone/frustum surface and volume; composite shapes (G14–G16, Extension).
• Pythagoras and trigonometry in 2D and 3D; sine/cosine rules and ½ab sinC (G19–G20, Extension; 1E 62.7°, 2E 102.6° and cos x = 8/11).
• Transformations including combined and fractional/negative enlargements (G21, Extension).
• Vector notation and geometry (G22; 2E CB ∥ MN proof).
• Matrix multiplication, identity, unit-square transformations and combinations (G23–G26, Extension; 1E matrices).

Boundary notes: NO formulae sheet in 9260. Lines marked "not sampled" (plans and elevations, parts of the solids programme) are specification originals; the specimens are Extension-tier only. Worked examples are adapted from the 2016 specimen pair or original.