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Constructions and Loci - Edexcel GCSE Higher Maths

Typically 3-5 marks per paper, usually one construction or one shaded region · Spec 1MA1

A construction is an accurate drawing made with compasses and a straight edge. A locus is the set of every point that obeys one rule, and the plural is loci. Edexcel keeps both at spec reference G2 on 1MA1: use the standard ruler and compass constructions (the perpendicular bisector of a line segment, a perpendicular to a given line from or at a given point, and the bisector of a given angle), use them to construct figures and solve loci problems, and know that the perpendicular distance from a point to a line is the shortest distance to that line. G2 is worded identically for Foundation and Higher, so it is crossover content rather than a Higher-only topic. Scale drawings and maps live next door at G15, and Edexcel regularly wraps the two references into a single question.

The exam question is a drawing, and the wording barely changes from series to series: 'Use ruler and compasses to construct the perpendicular bisector of AB. You must show all your construction lines', worth 2 marks. The loci version gives a scale drawing of a garden, a park or a harbour and asks you to shade the region that satisfies two conditions at once, worth 3-4 marks, with the same instruction about construction lines attached. Every 1MA1 paper lists a ruler graduated in centimetres and millimetres, a protractor and compasses on its front cover, and tracing paper is permitted, so the kit is assumed rather than optional. The topic can appear on Paper 1, 2 or 3; a calculator changes nothing about the drawing, though it earns its keep once a scale has to be converted or an area worked out.

The marks are for method, and the arcs are the method. On Edexcel schemes the first mark for a perpendicular bisector is for a pair of arcs of equal radius centred on each end of the line, so a candidate who finds the midpoint with a ruler and drops a line with a protractor produces a picture that looks right and scores nothing; rubbing the arcs out afterwards costs exactly the same. Lines are generally accepted within about 2 mm, which is how a blunt pencil and a loose compass hinge lose marks that the geometry never lost. Because this page cannot hand you a diagram, the practice below tests the parts that survive without one: which construction produces which locus, what a described set of arcs has actually built, and the lengths, angles and areas that fall out once the region is drawn.

Worked example

A map is drawn to a scale of 1 cm to 50 m. Two masts, A and B, are marked on the map. On the map, AB measures 9.6 cm. A receiver is to be placed at the point P, which is the same distance from A as it is from B and is 300 m from A. Work out the real distance from P to the midpoint of AB.
[4 marks]
  1. Convert the map distance into a real distance first: 9.6×50=4809.6 \times 50 = 480, so the masts are 480 m apart on the ground.
    M1: correct use of the scale, map length multiplied by 50 (dividing here is the standard slip)
  2. Every point the same distance from A as from B lies on the perpendicular bisector of AB. That line meets AB at the midpoint M and meets it at 90∘90^\circ, so AM=240AM = 240 m and angle AMP=90∘AMP = 90^\circ.
    M1: identifying the locus as the perpendicular bisector and pulling out the right angle at M
  3. Triangle AMP is right-angled at M with AP=300AP = 300 m as the hypotenuse, so PM2=3002−2402=90000−57600=32400PM^2 = 300^2 - 240^2 = 90000 - 57600 = 32400.
    M1: Pythagoras with 300 correctly placed as the hypotenuse, so the squares are subtracted
  4. PM=32400=180PM = \sqrt{32400} = 180. P is 180 m from the midpoint of AB.
    A1: 180 m; the answer is in metres because the scale was applied before any geometry

Practice questions

Everything below is original, written in Edexcel 1MA1 Higher style for a page with no diagrams, so each figure arrives in words: the points are named, every length carries its unit, and any right angle is stated. That rules out asking you to draw, which is the exam skill, so what is tested here is what the drawing rests on: matching a description to the construction that produces it, reading what a stated set of arcs has built, and finishing the calculation the locus sets up. Have a calculator to hand for the questions that ask for an area to 1 decimal place; the angle and scale questions are Paper 1 arithmetic.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [2 marks]
    An angle of 84∘84^\circ is bisected using ruler and compasses. Work out the size of each of the two angles produced, in degrees.
  2. Question 2 Warm-up [2 marks]
    A scale drawing of a hall uses a scale of 1 cm to 4 m. One wall of the hall is 18 m long. Work out the length of that wall on the scale drawing, in cm.
  3. Question 3 Exam pace [2 marks]
    A goat is tied by a 7 m rope to a post in the middle of a large flat field. The rope is pulled tight as the goat walks. Work out the area of ground the goat can reach. Give your answer in m2^2, correct to 1 decimal place.
  4. Question 4 Exam pace [1 mark]
    A question asks you to shade the region inside triangle ABC that is both less than 5 cm from B and closer to AB than to BC. Which pair of constructions gives the boundaries of that region?
    Answer options for question 4
  5. Question 5 Exam pace [1 mark]
    P and Q are two points 10 cm apart. Which of these is the set of all points that are closer to P than to Q?
    Answer options for question 5
  6. Question 6 Stretch [3 marks]
    Triangle PQR has a right angle at Q, with PQ = 5 cm and QR = 12 cm. Work out the shortest distance from Q to the line PR. Give your answer in cm, correct to 2 decimal places.

Common mistakes examiners see

  • Rubbing out the construction arcs so the page looks tidy, leaving one clean line where the perpendicular bisector should be.

    Leave every arc on the paper. Edexcel gives the first mark for the arcs, not for the line, so a bisector with no arcs is usually worth nothing even when it is in exactly the right place. Draw arcs long enough to cross clearly above and below the line, and never sharpen the picture up at the end.

  • Changing the compass width between the two centres, or setting it to less than half of AB, so the arcs from A and from B either cross off-centre or never cross at all.

    Open the compasses to a little over half the length of AB, then lock the hinge and do not touch it until both pairs of arcs are drawn. Equal radii are what force the crossing points to be the same distance from A and from B, which is what makes the line a bisector.

  • Drawing the locus of points 3 cm from a line segment PQ as two parallel lines and stopping, so the region has open ends and the shading leaks past P and Q.

    Treat the two end points as points in their own right. The finished locus is a racetrack: a straight line 3 cm from PQ on each side, capped by a semicircle of radius 3 cm centred on P and another centred on Q. Square corners are the giveaway that the ends were forgotten.

  • Using the wrong bisector: drawing the perpendicular bisector when the question says equidistant from two sides, or ruling a diagonal from corner to corner of a rectangle and calling it the locus equidistant from two edges.

    Two points give a perpendicular bisector; two lines give an angle bisector. Read the noun in the question before you pick up the compasses. In a rectangle the diagonal only happens to bisect the corner angle when the rectangle is a square, so construct the bisector properly rather than joining the corners.

  • Shading a region that satisfies only one of the conditions, or reading 'closer to A than to B' as 'on the perpendicular bisector' and shading nothing but the line.

    Draw each boundary first, then test one point you can see. If a point in the region really is nearer A and really is inside the arc, shade that region and label it R. The bisector itself is the equal case, so 'closer to A' means the whole side of it that contains A.

Frequently asked questions

Is constructions and loci on the Foundation paper as well as Higher?
Yes. G2 is crossover content and the specification wording is the same at both tiers, so a Foundation candidate meets the same three constructions and the same locus rules. What changes at Higher is the packaging: the region is defined by two or three conditions rather than one, the scale drawing has to be converted before or after the geometry, and the question may finish with Pythagoras or an area calculation rather than with the shading.
Do I need a pair of compasses for the Edexcel GCSE maths exam?
Yes. The front cover of every 1MA1 paper lists a ruler graduated in centimetres and millimetres, a protractor, compasses, a pen, an HB pencil and an eraser, and it adds that tracing paper may be used. A perpendicular bisector produced by measuring the midpoint and using a protractor will not score, because the mark scheme asks for arcs. Take a compass that holds its setting, and a spare if you have one.
Why do I have to show my construction lines?
Because the marks are for the method and the arcs are the only record of it. A question that says you must show all your construction lines is telling you where the first mark lives: a pair of equal arcs centred on each end of the line, or an arc centred on the vertex crossing both arms of the angle. The final line is then usually accepted within about 2 mm of the true position.
How many marks are constructions and loci questions worth on Edexcel Higher?
A single construction is normally 2 marks, and a loci question that asks for a region satisfying two conditions is usually 3-4 marks, so expect roughly 3-5 marks in a paper that carries the topic. Scale questions attached to the same diagram add a mark or two, either for converting a map distance to a real one or for writing a scale such as 1 cm to 25 m in the form 1 : k.

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