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Loci and construction

Free Printable Loci and construction PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 4)

Download free loci and construction worksheets (PDF), GCSE maths past paper questions and worked solutions for Loci and construction (4Z). This is a Grade 4 geometry topic from the GCSE questions by topic collection for Foundation and Higher tier crossover GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Corbett Maths, Maths Made Easy, Naiker Maths, Dr Austin, Tom Riley, 1st Class Maths, Metatutor, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our loci and construction revision notes and worked examples with diagrams, try real GCSE past paper loci and construction questions with step-by-step solutions, or download a PDF of all loci and construction worksheet links.

Free Loci and construction PDF Worksheets & Answers

Grade 4 • Topic 4Z
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Tom Riley
Loci and construction questions & answers
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Grade 4 topics
4A - Compound interest4B - Indices (index laws)
4C - Prime factors HCF and LCM4D - Real life graphs
4E - Inequalities on a number line4F - Solving inequalities
4G - Forming and solving equations4H - nth term of a linear sequence
4J - Expanding and factorising4K - Pythagoras theorem
4L - Angles in parallel lines4M - Angles in polygons
4N - Surface area4P - Volume of a cuboid
4Q - Volume of a prism4R - Bearings
4S - Cylinders4T - Plans and elevations
4U - Scatter graphs4V - Probability
4W - Averages from tables4X - Reverse mean
4Y - Area of a trapezium4Z - Loci and construction
GCSE Maths Grade 4 Revision Notes & Method

Guide to Loci and construction: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

A locus (plural: loci) is the path of points that obey a given distance rule. In construction and loci questions, never rub out your compass construction arcs — examiners award method marks for visible arcs!

Know the four standard loci: (1) Fixed distance from a point P → a circle around P; (2) Equidistant from two points A and B → the perpendicular bisector of line AB; (3) Equidistant from two intersecting lines → the angle bisector; (4) Fixed distance from a line segment → parallel lines with semicircular ends (stadium shape).

Example 1: Construct the perpendicular bisector of line segment AB (equidistant from A and B).

A B
1. Radius > 12AB.
2. Equal arcs from A & B above & below.
3. Join intersections.

Example 2: Garden ABCD has scale 1 cm = 2 m. Shade the region closer to A than B and within 6 m of C.

A B C D r = 3 cm (6 m)
1. Bisect AB (left side).
2. 6 ÷ 2 = 3 cm arc centred at C.
3. Shade overlap.
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Loci and construction Questions & Answers

Past Paper Example 1 OCR Jun 2017 6H Q2 7 marks
Full Paper (PDF) • Mark Scheme (PDF)
a(i) (i) The length of a path is 240 m. Work out the length, in centimetres, of the path on the map.
a(ii) (ii) The scale 1 cm represents 25 m can be written in the form 1 : k. Find the value of k.
b (b) The scale drawing represents a park.
(a) The length of a path is 240 m. Work out the length, in centimetres, of the path on the map. [1 mark]
The scale 1 cm represents 25 m can be written in the form 1 : k. Find the value of k. [1 mark]
(b) The scale drawing represents a park. [5 marks]
Scale: 1 cm represents 25 mBCAD
A new play area must be
- no more than 150 m from B
- closer to AD than to CD.
Construct and shade the region where the play area can be positioned. Show all your construction lines.
Worked Solution & Mark Scheme (7 marks)
Part (a) Solution:
240 ÷ 25 = 9.6 cm B1
Answer (a): 9.6 cm cm
Part Solution:
25 m = 2500 cm, so 1 : 2500 ⇒ k = 2500 B1
Answer : (ii) k = k = 2500
Part (b) Solution:
150 m ÷ 25 = 6 cm M4
Draw an arc of radius 6 cm centred at B, construct the angle bisector of ∠ ADC, and shade the region inside the arc and closer to AD A1
Scale: 1 cm represents 25 mBCAD
Answer (b): Arc of radius 6 cm centred at B, angle bisector of ∠ADC with construction arcs, and region shaded
Past Paper Example 2 AQA Jun 2019 2F Q20 / 2H Q5 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
Using ruler and compasses, show the region inside the grid that is
less than 4 cm from A
and
nearer to B than to C.
ABC
Label the region R.
Show all your construction lines.
Worked Solution & Mark Scheme (3 marks)
1. Draw an arc of a circle of radius 4 cm with centre A inside the grid. M1
2. Construct the perpendicular bisector of the line segment BC with compass arcs. M1
3. Identify and label region R inside the circular arc and on the side of the perpendicular bisector closer to B. A1
RABC
Answer: Answer Region R correctly constructed and shaded
Past Paper Example 3 AQA Nov 2017 1F Q21 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
(a) Joe wants to bisect angle BCD.
Here is his method.
Use a pair of compasses to draw arcs of the same radius from B and D.
Draw a straight line from C through the intersection of the arcs.
Write down the error in his method.
21 (b) Kay wants to show all the points 3 km from point P.
Scale: 1 cm represents 1 km
Here is her answer.
What is wrong with her answer?
21 (c) Here is a rectangle.
Using a pair of compasses and a straight edge, construct one line of symmetry.
Show clearly your construction arcs.
(a) Joe wants to bisect angle BCD. Here is his method.
Use a pair of compasses to draw arcs of the same radius from B and D.
Draw a straight line from C through the intersection of the arcs.
Write down the error in his method. [1 mark]
CBD
(b) Kay wants to show all the points 3 km from point P. Scale: 1 cm represents 1 km
Here is her answer. What is wrong with her answer? [1 mark]
Scale: 1 cm represents 1 km× P
(c) Here is a rectangle. [2 marks]
Using a pair of compasses and a straight edge, construct one line of symmetry.
Show clearly your construction arcs.
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
B and D are not the same distance from C
arcs must be drawn from points on CB and CD that are equidistant from C. B1
Answer (a): Answer B and D are not equidistant from C
Part (b) Solution:
The locus of points at a fixed distance from a point is a circle. B1
Scale: 1 cm represents 1 km× P
Answer (b): Answer It should be a circle of radius 3 cm, not a square
Part (c) Solution:
Construct the perpendicular bisector of either pair of opposite parallel sides. M1 A1
Answer (c): Answer Perpendicular bisector of opposite sides constructed with compass arcs

← 4Y Area of a trapezium

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