Geometry and measures

Explanation

• Geometry uses agreed words and symbols so a diagram communicates facts precisely. A point names a position; a line continues, while a line segment has endpoints.
• Vertices are corners and edges join vertices. Parallel lines never meet, whereas perpendicular lines meet at $90^\circ$.
• In $\angle ABC$, the middle letter $B$ is the vertex; in triangle $ABC$, side $AC$ is opposite $B$.
• Draw a written description one instruction at a time, label every required point and add conventional arrow, equality and right-angle marks.
• The examiner awards stated or marked relationships, never what a sketch merely appears to show.

Explanation

• Ruler-and-compass constructions must show the arcs that create the result. For a perpendicular bisector of $AB$, draw equal-radius arcs from $A$ and $B$ with radius greater than half of $AB$, then join their intersections.
• Every point on this line is equidistant from $A$ and $B$.
• An angle bisector is built from an arc centred at the vertex and equal arcs from the two cut points.
• Translate loci language into boundaries: a fixed distance from a point gives a circle; a fixed distance from a line gives two parallels.
• Perpendicular distance is the shortest distance to a line.

Explanation

• Use angle facts as a chain of justified steps. Angles at a point total $360^\circ$, angles on a straight line total $180^\circ$, and vertically opposite angles are equal.
• When a transversal crosses parallel lines, corresponding and alternate angles are equal; interior angles on the same side total $180^\circ$.
• A triangle totals $180^\circ$.
• Drawing diagonals from one vertex divides an $n$-gon into $n-2$ triangles, so its interior-angle sum is $(n-2)\times180^\circ$.
• In exam working, write the relevant equality or subtraction and name the angle fact, including the parallel lines that make it valid.

Explanation

• Classify a plane shape using properties that must hold, not how the drawing looks. A parallelogram has two pairs of parallel opposite sides; its opposite sides and angles are equal, and its diagonals bisect each other.
• A rectangle is a parallelogram with four right angles and equal diagonals. A rhombus has four equal sides and perpendicular diagonals, while a square is both a rectangle and a rhombus.
• A kite has two pairs of adjacent equal sides.
• An isosceles triangle has equal base angles.
• Examiners expect a property to be connected to a conclusion, especially in 'explain' and proof questions.

Explanation

• Congruent triangles have exactly the same size and shape, so corresponding sides and angles are equal. Prove congruence using SSS, SAS, ASA or RHS.
• SAS needs the angle included between the two known sides. RHS applies only to right-angled triangles and needs equal hypotenuses plus one equal shorter side.
• Mark or state the three matching facts, then write triangle names in corresponding order.
• Only after proving congruence may you use corresponding parts to establish another equality.
• AAA proves similarity rather than congruence, and SSA does not determine a unique triangle, so neither is a valid general congruence test.

Explanation

• Combine established geometry facts to derive a new result. Equal sides in an isosceles triangle face equal angles, and equal angles face equal sides.
• In a right-angled triangle, Pythagoras' theorem gives $a^2+b^2=c^2$, where $c$ is always the hypotenuse opposite the right angle.
• Angle facts, quadrilateral properties, similarity and congruence can then form a proof chain.
• Each line must follow from given information or a named result; a diagram's appearance is not evidence.
• For a 'prove' question, show the intermediate equality or congruence and finish with the exact conclusion requested.

Explanation

• Describe a transformation completely: a translation needs a column vector; a rotation needs centre, angle and direction; a reflection needs its mirror line; an enlargement needs centre and scale factor. Translation, rotation and reflection preserve every length and angle, so object and image are congruent.
• Enlargement preserves angles and multiplies every length from the centre by the scale factor. For centre $C$, use $\overrightarrow{CP'}=k\overrightarrow{CP}$.
• A fractional factor shrinks the shape.
• At Higher tier, a negative factor also puts every image point on the opposite side of the centre.
• Examiners require every defining detail, not just the transformation name.

Explanation

• Higher tier only. In a combination, apply transformations in the stated order because the image from one step becomes the object for the next.
• Two translations combine by adding their vectors. Two reflections in parallel lines produce a translation; two reflections in intersecting lines produce a rotation, with angle twice the angle between the mirror lines.
• Rotations, reflections and translations preserve lengths, angles, area and parallelism. One reflection reverses orientation; two reflections restore it.
• Coordinate rules are a reliable way to identify a single equivalent transformation.
• Reversing the order can change the result, so transformations do not generally commute.

Explanation

• Use circle vocabulary precisely. A radius joins the centre to the circumference.
• A diameter is a chord through the centre and has length $2r$. A chord joins two points on the circumference, while a tangent meets the circle at exactly one point.
• An arc is part of the circumference. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.
• The minor region uses the shorter arc and the major region uses the longer arc.
• In a labelled-diagram question, identify the defining boundaries rather than relying on a region's appearance.

Explanation

• Higher tier only.
• Standard circle theorems connect angles, radii, tangents and chords.
• The angle at the centre is twice the angle at the circumference standing on the same arc; angles in the same segment are equal; an angle in a semicircle is $90^\circ$; and opposite angles in a cyclic quadrilateral total $180^\circ$.
• A radius is perpendicular to a tangent at the contact point, tangents from one external point are equal, and the alternate segment theorem equates the tangent-chord angle with the angle in the opposite segment.
• A proof must identify the relevant common arc or chord and name each theorem used.

Explanation

• Coordinate geometry turns a diagram into exact calculations. The midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.
• Horizontal and vertical coordinate changes form a right triangle, so distance is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
• Coordinate differences can prove equal side lengths and identify horizontal or vertical lines; areas can be found from base and perpendicular height or by enclosing and subtracting simpler shapes.
• Keep brackets when subtracting negative coordinates and show the substitution.
• The axes and calculated differences, not the sketch's appearance, supply the evidence the examiner needs.

Explanation

• Describe a solid by its flat faces, curved surfaces, edges and vertices. A face is flat; an edge is where surfaces meet; a vertex is where edges meet.
• An $n$-sided prism has two congruent polygonal ends and $n$ rectangular side faces, so it has $n+2$ faces, $3n$ edges and $2n$ vertices.
• A pyramid has one polygonal base and triangular faces meeting at an apex.
• A cylinder has two circular faces and one curved surface but no vertices; a cone has one circular face, one curved surface and one vertex; a sphere has only one curved surface.
• Count hidden features once.

Explanation

• A plan is the view from directly above. A front or side elevation is an orthographic view from the named horizontal direction, drawn without perspective.
• Corresponding widths align between the plan and front elevation; depths align between the plan and side elevation. For stacks of unit cubes, a numbered plan records the height in each occupied cell.
• An elevation shows, for each viewing column, the greatest height visible along the line of sight.
• Use true lengths in the viewing plane and include changes of outline, but do not add hidden internal edges unless requested.
• Examiners compare dimensions and visible column heights, not artistic realism.

Explanation

• Choose standard units that match the quantity: length uses mm, cm, m or km; area uses square units; volume uses cubic units; capacity uses ml or litres; mass uses g or kg; and time uses seconds, minutes or hours.
• Convert all measurements to compatible units before calculating.
• A linear conversion factor must be squared for area and cubed for volume: because $1\text{ m}=100\text{ cm}$, $1\text{ m}^2=10\,000\text{ cm}^2$ and $1\text{ m}^3=1\,000\,000\text{ cm}^3$.
• Examiners expect a numerical value with the correct unit and sensible rounding for the context, especially money and time.

Explanation

• Measure only from an accurate scale drawing, using a ruler or protractor to a precision justified by the scale. For scale $1:n$, one unit on the drawing represents $n$ of the same units in reality, so multiply drawing lengths by $n$ and then convert units.
• A bearing is the clockwise angle from north at the starting point and is written with three figures, such as $052^\circ$.
• Draw a north line at the point where the journey starts.
• Reverse bearings differ by $180^\circ$.
• Examiners expect the scale working, the correct direction of measurement and a three-figure bearing.

Explanation

• Use $A=\frac12bh$ for a triangle, $A=bh$ for a parallelogram and $A=\frac12(a+b)h$ for a trapezium, where $h$ is perpendicular to the base or parallel sides.
• A right prism has a constant cross-section, so $V=\text{cross-sectional area}\times\text{length}$.
• A cylinder is a right prism with circular cross-section, giving $V=\pi r^2h$.
• Mark the cross-section first, calculate its area with square units, then multiply by the prism length to obtain cubic units.
• Examiners expect correct substitution and units; a sloping side is not a perpendicular height unless a right angle establishes it.

Explanation

• For a circle, use $C=2\pi r=\pi d$ and $A=\pi r^2$. Composite areas add included regions and subtract cut-outs; composite perimeters include only the outside boundary.
• A cone or pyramid has volume $V=\frac13\times\text{base area}\times\text{perpendicular height}$. A cone's curved surface area is $\pi rl$, where $l$ is slant height.
• A sphere has surface area $4\pi r^2$ and volume $\frac43\pi r^3$.
• For a composite solid, divide it into recognised solids and exclude internal joined faces from surface area.
• Examiners expect radius, perpendicular height and slant height to be distinguished correctly.