3.4 Geometry and measures

3.4.1 Properties and constructions

G1

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use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries use the standard conventions for labelling and referring to the sides and angles of triangles draw diagrams from written description

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use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries

use the standard conventions for labelling and referring to the sides and angles of triangles

draw diagrams from written description

G2

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	use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle) use these to construct given figures and solve loci problems know that the perpendicular distance from a point to a line is the shortest distance to the line

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use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle)

use these to construct given figures and solve loci problems

know that the perpendicular distance from a point to a line is the shortest distance to the line

Notes: including constructing an angle of 60°.

G3

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apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles understand and use alternate and corresponding angles on parallel lines derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)

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apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles

understand and use alternate and corresponding angles on parallel lines

derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)

Notes: colloquial terms such as Z angles are not acceptable and should not be used.

G4

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derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus and triangles and other plane figures using appropriate language

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derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus

and triangles and other plane figures using appropriate language

Notes: including knowing names and properties of isosceles, equilateral, scalene, right-angled, acute-angled, obtuse-angled triangles. Including knowing names and using the polygons: pentagon, hexagon, octagon and decagon.

G5

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	use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)

G6

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	apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

G7

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identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement	including fractional scale factors	including negative scale factors

G8

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		describe the changes and invariance achieved by combinations of rotations, reflections and translations

Notes: including using column vector notation for translations. See also G24

G9

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identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference	including: tangent, arc, sector and segment

G10

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		apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results

Notes: including angle subtended by an arc at the centre is equal to twice the angle subtended at any point on the circumference, angle subtended at the circumference by a semicircle is 90°, angles in the same segment are equal, opposite angles in a cyclic quadrilateral sum to 180°, tangent at any point on a circle is perpendicular to the radius at that point, tangents from an external point are equal in length, the perpendicular from the centre to a chord bisects the chord, alternate segment theorem.

G11

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solve geometrical problems on coordinate axes

G12

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identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres

G13

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interpret plans and elevations of 3D shapes	construct and interpret plans and elevations of 3D shapes

3.4.2 Mensuration and calculation

G14

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use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money etc.)

G15

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measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings

Notes: including the eight compass point bearings and three-figure bearings.

G16

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know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)

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know and apply formulae to calculate: area of triangles, parallelograms, trapezia;

volume of cuboids and other right prisms (including cylinders)

G17

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know the formulae: circumference of a circle = $2 π r = π d$ area of a circle = $π r^{2}$ calculate perimeters of 2D shapes, including circles areas of circles and composite shapes	surface area and volume of spheres, pyramids, cones and composite solids

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know the formulae: circumference of a circle = $2 π r = π d$

area of a circle = $π r^{2}$

calculate perimeters of 2D shapes, including circles

areas of circles and composite shapes

surface area and volume of spheres, pyramids, cones and composite solids

Notes: including frustums.

Solutions in terms of $π$ may be asked for. See also N8, G18

G18

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	calculate arc lengths, angles and areas of sectors of circles

Notes: see also N8, G17

G19

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	apply the concepts of congruence and similarity, including the relationships between lengths in similar figures	including the relationships between lengths, areas and volumes in similar figures

Notes: see also R12

G20

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	know the formulae for: Pythagoras’ theorem, $a^{2} + b^{2} = c^{2}$ and the trigonometric ratios, $\sin θ = \frac{opposite}{hypotenuse}$ , $\cos θ = \frac{adjacent}{hypotenuse}$ and $\tan θ = \frac{opposite}{adjacent}$
	apply them to find angles and lengths in right-angled triangles in two dimensional figures	apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures

Notes: see also R12

G21

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	know the exact values of $sin θ$ and $cos θ$ for $θ$ = 0°, 30°, 45° , 60° and 90° know the exact value of $tan θ$ for $θ$ = 0°, 30°, 45° , 60°

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know the exact values of $sin θ$ and $cos θ$ for $θ$ = 0°, 30°, 45° , 60° and 90°

know the exact value of $tan θ$ for $θ$ = 0°, 30°, 45° , 60°

Notes: see also A12

G22

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		know and apply the sine rule, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ and cosine rule, $a^{2} = b^{2} + c^{2} - 2 b c \cos A$ to find unknown lengths and angles

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know and apply the sine rule, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

and cosine rule,

$a^{2} = b^{2} + c^{2} - 2 b c \cos A$

to find unknown lengths and angles

G23

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		know and apply $A r e a = \frac{1}{2} a b \sin C$ to calculate the area, sides or angles of any triangle

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know and apply

$A r e a = \frac{1}{2} a b \sin C$

to calculate the area, sides or angles of any triangle

3.4.3 Vectors

G24

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describe translations as 2D vectors

Notes: see also G8

G25

Basic foundation content	Additional foundation content	Higher content only
	apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors	use vectors to construct geometric arguments and proofs

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3.3 Ratio, proportion and rates of change

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3.5 Probability