Properties and constructions — AQA GCSE Mathematics

Properties and constructions in AQA GCSE Mathematics covers the accurate drawing of triangles, angle bisectors, perpendicular bisectors, loci, and scale diagrams using a ruler, compass, and protractor. You must know the standard ruler-and-compass constructions and be able to describe regions bounded by loci, often linking them to real-world contexts.

Properties and constructions is a practical geometry topic in AQA GCSE Mathematics where you use a ruler, compass, and protractor to accurately draw shapes and loci. You will learn to construct triangles given specific side lengths and angles, bisect lines and angles, and draw perpendiculars from a point to a line. These skills are essential for solving real-world problems involving boundaries and regions.

This topic also introduces loci, which are sets of points that satisfy a given condition, such as being a fixed distance from a point or line. You will need to describe and shade regions defined by multiple loci, often in context. Mastery of constructions and loci is crucial for higher-tier questions and provides a foundation for further study in geometry and trigonometry.

• →Constructing triangles using SSS, SAS, ASA, and RHS criteria with a ruler, compass, and protractor.
• →Constructing the perpendicular bisector of a line segment and the bisector of an angle using compasses only.
• →Drawing perpendiculars from a point to a line and from a point on a line.
• →Understanding loci: the locus of points at a fixed distance from a point is a circle; from a line segment is a pair of parallel lines and semicircles at the ends; equidistant from two points is the perpendicular bisector; equidistant from two lines is the angle bisector.
• →Solving problems involving scale drawings and regions defined by multiple loci, including shading the required region.

• A diagram that carries every fact in the description, with equal sides shown by dashes, parallel sides by arrowheads, and perpendicular lines by a right-angle symbol.
• Naming an angle with three letters in the correct order, vertex in the middle, so the angle you mean is not ambiguous.
• Correctly identifying vertices and edges of a given polygon.
• The correct number of lines of symmetry or the correct order of rotation symmetry.
• Construction arcs left visible; a correct line without arcs scores no method marks.
• Using the correct construction to create a figure, such as a triangle with given side lengths (SSS).
• Each locus boundary drawn correctly using the appropriate construction.
• A final region correctly identified and shaded, satisfying all given conditions.
• For a perpendicular at a given point on a line, arcs of equal radius are drawn on the line either side of the point, then equal arcs from those intersections meet above or below the point.
• A correct angle fact stated or clearly used, such as 'alternate angles are equal'.
• A clear reason given when requested, naming the geometric property used.
• Showing the derivation of the triangle angle sum using parallel lines.
• Correctly calculating the interior angle sum of a polygon using (n − 2) × 180°.
• Correctly stating a definition for a shape, such as a kite having two pairs of adjacent equal-length sides, or a regular polygon having all sides and interior angles equal.
• Deriving a property from a definition, for example using alternate and co-interior angle facts to show why opposite angles in a parallelogram are equal.
• Identifying a triangle (e.g. isosceles) from given information and using its properties (e.g. equal base angles) to find a missing angle.
• Distinguishing between shapes, for example explaining why a rhombus is not necessarily a square, or why a square is both a rectangle and a rhombus.
• Using appropriate language such as 'adjacent', 'parallel', 'bisect' and 'perpendicular' when describing a shape, including regular polygons and circles.
• each pair of equal sides or equal angles identified together with its reason, such as a shared side or a given equal angle
• naming the correct condition once three matching facts are in place
• the concluding statement that the triangles are congruent, since the deduction is part of what is being marked
• writing the vertices in matching order, so the pairs claimed equal are the pairs that actually correspond
• Each logical step in a proof must be justified with a correct geometric reason (e.g., 'alternate angles are equal').
• Correctly identifying and applying the conditions for triangle congruence (SSS, SAS, ASA, RHS).
• Using properties of quadrilaterals as reasons, for example that opposite sides of a parallelogram are parallel.
• A concluding statement that summarises what has been proven, directly addressing the question.
• naming the single transformation correctly
• each supporting detail: centre, angle and direction for a rotation, the equation of the mirror line for a reflection, the vector for a translation, the centre and scale factor for an enlargement
• an image in the correct position and a further mark for the correct size and orientation, so a partly correct image can still score
• saying whether the object and image are congruent or similar when the question asks for that comparison
• the correct scale factor written as a fraction, found as an image length divided by the matching object length
• multiplying each distance from the centre by the scale factor rather than subtracting from it
• an image of the correct size, and a further mark for it being in the correct position relative to the centre
• a description naming the transformation as an enlargement and giving both the centre and the fractional scale factor
• placing image vertices on the opposite side of the centre from the object
• distances from the centre scaled by the size of the scale factor, with the sign handled by the change of direction
• a fully correct image, the right size and turned through a half turn relative to the object
• a description giving both the centre of enlargement and the negative scale factor, sign included
• a correct intermediate image after the first transformation, credited even if the second transformation then goes wrong
• naming the single transformation equivalent to the pair
• the full detail of that single transformation, such as the centre together with the angle and direction
• correctly identifying the invariant points or invariant lines, with a reason for why they do not move
• the correct name of a labelled part of a circle
• doubling a radius to reach the diameter, or halving a diameter to reach the radius, when the question gives one and the formula needs the other
• a correct substitution into circumference = 2πr, even when the arithmetic that follows is wrong
• the final value rounded as instructed and given with the correct units
• correctly naming or labelling a region or line on a circle diagram, for example identifying the shaded region as a segment because a chord bounds it
• the fraction of the circle a sector represents, found as the angle at the centre divided by 360°
• using the right angle between a tangent and the radius at the point of contact as the opening fact in a problem
• a correct arc length or sector area from a correct fraction; where the fraction is wrong, credit depends on the wording of the particular question and its mark scheme, so do not assume follow-through
• A correct calculation or deduction as part of a multi-step problem, with a clear geometric reason stated.
• Correctly identifying and stating the full name of a circle theorem used to justify a step.
• For 'prove' questions, a complete and logical chain of reasoning from the given information to the required conclusion.
• Clear algebraic formulation of the problem, for example, letting an unknown angle be 'x' and forming an equation.
• the correct differences in the x direction and the y direction before any formula is applied
• a correct substitution into a² + b² = c² when a length between two points is needed
• averaging both coordinates when finding a midpoint rather than only one of them
• a correct gradient, and correct use of the negative reciprocal when perpendicularity is being tested; the number of marks awarded depends on the particular question and its mark scheme
• the correct count of faces, edges or vertices as asked for, including the two circular edges of a cylinder and the single circular edge of a cone
• treating the apex of a cone as a vertex, so a cone has one vertex, while a cylinder has none
• correctly describing the cross-section when a prism has to be identified or named
• distinguishing a flat face from a curved surface, since a cylinder has two of one and one of the other
• a net with every face present and in the right shape, where a net is asked for
• identifying which view is which, matching the plan to the footprint of the solid
• a solid that agrees with two of the three views when the third has been misread
• the correct dimensions or the correct number of cubes read off the views
• naming or describing the solid itself, not only for repeating what each view shows
• each view drawn with the correct outline shape
• correct dimensions on each view, taken from the same scale as the other views
• internal lines shown wherever the height or depth of the solid changes
• labelling which drawing is the plan and which are the front and side elevations

• 💡Mark given facts onto the diagram as you read: dashes for equal sides, arrows for parallel lines, squares for right angles. This makes the geometry clearer.
• 💡If no diagram is provided, draw one. A quick sketch with given lengths and angles helps visualise the problem and prevent errors.
• 💡Use a sharp pencil and ensure your compass hinge is tight to prevent the radius from changing during construction, which leads to inaccurate results.
• 💡When constructing a figure like a triangle, start by drawing one side to the correct length, then use constructions to find the location of the third vertex.
• 💡For a perpendicular at a point on a line, first mark equal distances along the line from the point, then construct the perpendicular bisector of that new segment.
• 💡When a question asks for reasons, state the full geometric fact (e.g., 'angles on a straight line sum to 180°'). This demonstrates your understanding and is good practice.
• 💡If a diagram is labelled 'not drawn accurately', do not use a protractor. Every angle must be calculated using geometric properties.
• 💡For 'describe' questions, structure your answer around sides, angles, diagonals and symmetry.
• 💡When a shape is described in words, sketch it and mark on the properties you know (e.g. parallel lines, equal sides).
• 💡Use definitions to derive properties rather than quoting them: for example, use alternate and co-interior angle facts to explain why opposite angles in a parallelogram are equal.
• 💡Set the answer out as three facts in a column, each with its reason, then the condition, then the conclusion. That layout matches how the marks are handed out.
• 💡Look hard for a shared side and for angles equal because they are vertically opposite or alternate; those are the facts questions hide rather than state.
• 💡Redraw the two triangles separately, the same way round, if the diagram overlaps them. Corresponding parts become much easier to see.
• 💡Marks are awarded for both mathematical statements and the geometric reasons that justify them. Always provide a reason for each step in a proof.
• 💡In 'show that' or 'prove' questions, the answer is given. You must construct a rigorous argument to get there; you cannot use the answer as part of your working.
• 💡For an enlargement, count squares from the centre to one vertex, multiply by the scale factor and count out again. Repeat for every vertex instead of guessing the shape.
• 💡If your description needs the word then, you have found two transformations and the question wants one. Look again for the single one that does the whole job.
• 💡Tracing paper is allowed in the exam and turns an awkward rotation into a physical turn. Ask for it if you do not have it.
• 💡Check the finished drawing by comparing one pair of corresponding sides; dividing image length by object length must return the scale factor you used.
• 💡Say before you draw whether you expect the image to be bigger or smaller. A scale factor below 1 shrinks the shape, above 1 grows it.
• 💡Mark the centre of enlargement clearly with a cross, because every measurement in the question is taken from it.
• 💡Rule a faint straight line from an object vertex through the centre and keep going. The matching image vertex lies on that line, past the centre.
• 💡Before committing to the drawing, check the image looks upside down compared with the object. A negative scale factor always turns the shape round.
• 💡Do the counting in two steps, across then up, and reverse both directions. Reversing only one of them is the usual slip.
• 💡Label the first image and the second image with different letters. The comparison that earns the marks is object against final image, and mixing the two images up loses everything after that point.
• 💡Find invariant points by asking which points the transformation leaves exactly where they are; the mirror line and the centre of rotation are the usual answers.
• 💡Sketch the object and the final image on their own, without the middle stage, before you try to name the single transformation.
• 💡Read the given number twice to decide whether it is a radius or a diameter. That single check recovers most of the marks lost on this content.
• 💡Leave the answer in terms of π when an exact value is asked for, and only use the calculator button when a decimal is wanted.
• 💡Write the formula down before substituting. It costs one line and it protects the method mark if the numbers go wrong.
• 💡Draw the two radii onto the diagram before starting a sector question. Seeing the slice makes the fraction of the circle obvious.
• 💡Whenever a tangent appears, mark the right angle where it meets the radius. That right angle is what makes the rest of the working possible.
• 💡Check whether the question wants the curved arc only or the whole perimeter, which also includes the two radii.
• 💡Always mark equal radii on the diagram. This creates isosceles triangles, and their equal base angles are often the key to solving the problem.
• 💡For a 'prove' question, start by identifying the given facts and the goal. Then, build a step-by-step argument. For example: 'Let angle ABC = x. Then angle AOC = 2x (reason: angle at centre is twice angle at circumference)'.
• 💡Sketch the points on axes even when the question gives only coordinates. The picture tells you at once whether a length, a gradient or a midpoint is wanted.
• 💡To show two sides are equal, compare the squared lengths. You avoid the surds entirely and the comparison is still valid.
• 💡Label the two points before substituting, so the differences are taken in a consistent order across and up.
• 💡Count edges in groups: all the ones on the top, then all on the bottom, then the uprights. Counting at random on a drawing is where the marks disappear.
• 💡A prism is named by its cross-section, so find the two identical parallel faces before you name it.
• 💡When counting on a picture, tick each face, edge or vertex with your pencil as you go so nothing is counted twice.
• 💡Turn the page so you are genuinely looking from the direction asked about. Drawing what you imagine instead of what you see is where the errors start.
• 💡Count squares on the grid rather than estimating; plans and elevations are drawn to scale, so the counts have to match between views.
• 💡Check your rebuilt solid against all three views before answering, not only against the one that was easiest to read.
• 💡Line the front elevation up directly below the plan. Their widths must be the same, and lining them up makes any mistake visible at a glance.
• 💡Draw only the outline plus the lines where the surface changes level. Nothing is drawn receding into the page.
• 💡Work in pencil and count the squares for each edge before ruling it, so the three drawings stay in step with one another.
• 💡Always use a sharp pencil and leave all construction arcs visible. Marks are awarded for correct construction method, not just the final line.
• 💡When shading a region defined by loci, use a clear pattern or colour and ensure the boundaries are accurately drawn. Label the region if required.
• 💡For scale drawing questions, always state the scale you are using and measure accurately. Check that your final answer is reasonable in the context of the problem.

• Writing 'angle B' when multiple angles meet at vertex B, creating ambiguity. The correction is to use three-letter notation, e.g., ∠ABC.
• Confusing parallel and perpendicular lines, or forgetting to use the correct notation on a diagram. The correction is to mark parallel lines with arrows and perpendicular lines with a square symbol.
• Calling a shape regular just because its sides are equal (e.g., a rhombus), forgetting the angles must also be equal. A regular polygon must have both equal sides and equal angles.
• Rubbing out construction arcs, removing evidence of the method used. The correction is to leave all arcs on the diagram as they are part of the working.
• Using a ruler or protractor to measure when the command word is 'construct'. The correction is to use only a ruler for straight lines and a pair of compasses for arcs.
• Forgetting to open the compasses to more than half the line segment length for a perpendicular bisector, so the arcs do not intersect. The correction is to visually check the compass width is clearly over halfway.
• Giving only one angle bisector as the locus equidistant from two intersecting lines. The correction is to draw both bisectors, since points on either line are equidistant from the two original lines.
• Confusing alternate and co-interior angles. The correction is to remember alternate angles form a 'Z' shape and are equal, while co-interior form a 'C' or 'U' shape and sum to 180°.
• Applying parallel line rules (alternate, corresponding) to lines that are not marked as parallel. The correction is to check for arrow symbols on the lines before using these rules.
• Calculating the exterior angle (360°/n) but giving it as the answer for the interior angle. The correction is to subtract the exterior angle from 180° to find the interior angle.
• Confusing a trapezium (one pair of parallel sides) with a parallelogram (two pairs of parallel sides). Correction: check how many pairs of opposite sides are parallel before naming the shape.
• Assuming both diagonals of a kite always bisect each other. Correction: In a kite, one diagonal is the perpendicular bisector of the other. Both diagonals only bisect each other in the special case where the kite is also a rhombus.
• Assuming diagonals of a parallelogram are perpendicular. Correction: This is only true for a rhombus or a square, which are special types of parallelogram.
• Claiming a kite can never be a parallelogram. Correction: A rhombus satisfies the definition of a kite (two pairs of adjacent equal sides) and is also a parallelogram, so the shapes are not mutually exclusive.
• using two sides and an angle that is not between them and calling the argument a valid condition
• claiming congruence from three equal angles, which only shows that the triangles are similar
• missing the side that two overlapping triangles share, so only two facts are found and the argument stops short
• listing the vertices in the wrong order, which makes the correspondence between the triangles wrong even when the facts are right
• Assuming the result you are trying to prove. Correction: You cannot use a property you are asked to prove as one of your reasons.
• Stating incorrect or incomplete conditions for congruence. Correction: AAA and ASS are not conditions for congruence; AAA proves similarity.
• Confusing the properties of different quadrilaterals. Correction: For example, stating that the diagonals of a parallelogram are equal is incorrect (this is only true for rectangles and squares).
• describing one transformation as two, such as a reflection followed by a translation, when a single transformation was asked for
• giving a rotation without its direction, so an angle of 90° could be either way round
• describing the mirror line in words, such as the line up the side, instead of giving its equation, for example x = 0
• reading the enlargement scale factor as the object length divided by the image length, which inverts it
• believing a shape cannot be enlarged by a fraction, and so naming the transformation as something other than an enlargement
• inverting the scale factor, so an image half the size of the object is described with a scale factor of 2
• halving the coordinates of each vertex when the centre of enlargement is not at the origin
• measuring the reduced distance from the object rather than from the centre of enlargement
• drawing the image on the same side of the centre as the object, which is what a positive scale factor does
• flipping the shape as though it were reflected, so the image is a mirror image rather than a half turn
• leaving the minus sign out when writing the scale factor in a description
• counting the reversed distance from a vertex of the object instead of from the centre of enlargement
• applying the two transformations in the wrong order; a rotation followed by a reflection is generally not the same as that reflection followed by the rotation
• describing both steps separately when the question asks for the single transformation equivalent to both
• calling the whole shape invariant when only the points lying on the mirror line are
• thinking an invariant point only has to stay inside the shape, when it has to map exactly onto itself
• putting the diameter into circumference = 2πr in place of the radius, which doubles the answer
• calling any straight line across a circle a diameter when it does not pass through the centre; such a line is a chord
• giving a circumference in square units, which belong to area rather than to length
• rounding π to 3 at the start, so the final answer misses the accuracy the question asked for
• confusing a sector with a segment; a sector is bounded by two radii, a segment by a chord
• using the length of the chord in place of the arc when working round the curved edge of a sector
• assuming a tangent passes through the centre, when a tangent never reaches the centre at all
• taking the fraction of the circle from an angle inside a triangle rather than the angle at the centre
• Assuming a quadrilateral is cyclic when one or more of its vertices are not on the circumference.
• In proofs, making an assumption without justification, such as assuming a line is a diameter or that a shape is a kite.
• Confusing the alternate segment theorem: the angle between the tangent and chord should be equated to the angle in the *opposite* segment, not the adjacent one.
• subtracting the coordinates to find a midpoint instead of adding them and halving
• swapping the order of x and y, so a point is plotted or read the wrong way round
• finding the difference in one direction only and calling that the distance between the two points
• taking the negative of a gradient rather than its negative reciprocal when testing for perpendicular lines
• counting only the faces visible in the drawing, so the hidden ones behind the solid are missed
• calling the surface of a sphere a face or giving a sphere edges, when it has one curved surface and no edges or vertices
• describing every solid with a pointed top as a pyramid, when a cone has a circular base and a curved surface instead of flat triangles
• naming a prism after the rectangle facing you rather than after its cross-section
• saying a cone has no vertices, when its apex is a vertex, or saying a cylinder has vertices, when it has none
• confusing the plan with the front elevation, so the solid is rebuilt as though seen from the wrong direction
• adding perspective or shading, so the drawing shows depth instead of the flat outline seen square on
• ignoring a line inside a view, so a stepped solid is read as a plain box
• judging sizes by eye when the views are drawn on squared paper and the squares can be counted
• drawing a three dimensional picture of the solid rather than a flat view from one direction
• leaving out the internal line where a step changes level, so two different solids end up with identical drawings
• drawing the side elevation using the width of the front face instead of the depth of the solid
• changing scale between the plan and the elevations, so widths that should match no longer do
• Students often think that the locus of points equidistant from two points is a straight line parallel to the line joining them, but it is actually the perpendicular bisector of the line segment joining the two points.
• When constructing an angle bisector, students may incorrectly place the compass at the ends of the angle arms instead of at the vertex, leading to an inaccurate bisector.
• Students sometimes forget that the locus of points at a fixed distance from a line segment includes semicircles at the endpoints, not just parallel lines.

1. 1Week 1: Revise the basic constructions: perpendicular bisector, angle bisector, and perpendicular from a point to a line. Practice each construction at least five times until you can do them accurately without instructions.
2. 2Week 1: Learn the four standard loci: circle, parallel lines with semicircles, perpendicular bisector, and angle bisector. For each, draw diagrams and write a definition in your own words.
3. 3Week 2: Practice constructing triangles given different combinations of sides and angles (SSS, SAS, ASA, RHS). Use a scale of 1 cm = 1 unit for scale drawing questions.
4. 4Week 2: Work through exam-style questions that involve shading regions defined by two or more loci. Focus on interpreting the conditions and combining the loci correctly.
5. 5Week 2: Complete a past paper question on properties and constructions under timed conditions. Review your answers, checking that all construction arcs are visible and regions are correctly shaded.

• 📋Construction questions: 'Construct the perpendicular bisector of line AB' or 'Construct the bisector of angle XYZ'. You must show all construction arcs and use a ruler for final lines.
• 📋Scale drawing and loci: 'A treasure is buried within 5 m of point A and equidistant from points B and C. Shade the region where the treasure could be.' You need to draw the loci accurately and shade the intersection.
• 📋Triangle construction: 'Construct a triangle with sides 7 cm, 5 cm, and 4 cm. Measure the largest angle.' Use a compass and ruler, and measure accurately.
• 📋Problem-solving with loci: 'A dog is tied to a post by a 3 m leash. The post is 2 m from a wall. Draw and shade the region the dog can reach.' This requires combining a circle and a straight line boundary.