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Loci and construction Quiz

Grade 4 • 4Z • 10 Questions (35 Marks)

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Loci and construction — Notes & Worked Examples Show / Hide

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.
Question 1
[2 marks]
Use ruler and compasses to construct the bisector of angle ABC.
You must show your construction lines.
ABC
Worked Solution & Answer
Arcs on AB and BC, intersecting arcs, straight line through B M1 A1
ABC
Answer: Angle bisector ray drawn with compass construction arcs
Question Paper (PDF) • Mark Scheme (PDF)
Question 2 AQA Nov 2020 3F Q22 / 3H Q6
[2 marks]
ABCD represents the plan of a field.
There is a path across the field that
starts at B
is the same distance from BA and BC.
ABCD
Using ruler and compasses, show the position of the path.
Worked Solution & Answer
The path is equidistant from the two intersecting lines BA and BC, which is the angle bisector of angle ABC M1
Construct using compass arcs centered at B and intersecting arcs inside the angle. A1
ABCDpath
Answer: Angle bisector of angle ABC drawn with construction arcs
Question Paper (PDF) • Mark Scheme (PDF)
Question 3 IGCSE Jun 2025 2F Q21 / 2H Q4
[2 marks]
AB
Using ruler and compasses only, construct the perpendicular bisector of the line AB
Show all your construction lines.
Worked Solution & Answer
Draw arcs of equal radius (> ½ AB) centred at A and B M1
join their intersections A1
AB
Answer: Perpendicular bisector line drawn with two pairs of intersecting construction arcs
Question Paper (PDF) • Mark Scheme (PDF)
Question 4 IGCSE Nov 2024 1F Q19 / 1H Q2
[2 marks]
ABC
Use ruler and compasses only to construct the bisector of angle ABC
You must show all your construction lines.
Worked Solution & Answer
Draw an arc centred at B crossing BA and BC M1
draw intersecting arcs of equal radius from those two points and join B through their intersection. A1
ABC
Answer: Correct angle bisector constructed with compass arcs shown
Question Paper (PDF) • Mark Scheme (PDF)
Question 5 AQA Jun 2019 2F Q20 / 2H Q5
[3 marks]
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 & Answer
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
Question Paper (PDF) • Mark Scheme (PDF)
Question 6 AQA Nov 2024 1H Q11
[3 marks]
Construct the region that lies within 4 cm of the line.
Worked Solution & Answer
Draw parallel straight lines 4 cm from the segment and semicircular arcs of radius 4 cm centred at each endpoint. M2 A1
Answer: Two parallel lines 4 cm above and below the line segment joined by semicircular arcs of radius 4 cm at each end
Question Paper (PDF) • Mark Scheme (PDF)
Question 7 AQA Nov 2017 1F Q21
[4 marks]
(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
Answer
(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
Answer
(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.
Answer
Worked Solution & Answer
(a)
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
(b)
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
(c)
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
Question Paper (PDF) • Mark Scheme (PDF)
Question 8 OCR Nov 2018 4H Q7
[5 marks]
The diagram shows triangle ABC.
ABC
(a) Construct the bisector of angle BAC. [2 marks]
(b) Construct the perpendicular bisector of AC. [2 marks]
(c) Shade the region inside triangle ABC that is
• nearer to AC than to AB
• nearer to A than to C. [1 mark]
Worked Solution & Answer
(a)
Draw arcs from A on AB and AC, then intersecting arcs to bisect angle BAC. M1 A1
Answer (a): Angle bisector of angle BAC constructed with supporting compass arcs
(b)
Draw intersecting arcs of equal radius (> 1/2 AC) from A and C and join their intersections. M1 A1
Answer (b): Perpendicular bisector of AC constructed with supporting compass arcs
(c)
Shade the region below the angle bisector of BAC and to the left of the perpendicular bisector of AC. B1
Answer (c): Region bounded by AC, angle bisector of BAC and perpendicular bisector of AC shaded
Question Paper (PDF) • Mark Scheme (PDF)
Question 9 OCR Nov 2020 4H Q5
[5 marks]
ABCD is a quadrilateral.
ABCD
(a) Construct the bisector of angle ABC.
Show all your construction lines. [2 marks]
(b) Construct the perpendicular bisector of BC.
Show all your construction lines. [2 marks]
(c) Shade the region which is
• nearer to BC than to AB
and
• nearer to B than to C. [1 mark]
Worked Solution & Answer
(a)
Draw arc centred at B cutting BA and BC, then intersecting arcs of equal radius from those two points. M1 A1
Answer (a): Ruled angle bisector of angle ABC drawn with two pairs of construction arcs
(b)
Draw arcs of equal radius (> ½BC) centred at B and C above and below BC and join their intersections. M1 A1
Answer (b): Ruled perpendicular bisector of BC drawn with two pairs of intersecting arcs
(c)
Shade the triangular region bounded by BC, the angle bisector of B, and the perpendicular bisector of BC. B1
Answer (c): Region between BC, the angle bisector of ABC, and the perpendicular bisector of BC shaded
Question Paper (PDF) • Mark Scheme (PDF)
Question 10 OCR Jun 2017 6H Q2
[7 marks]
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]
cm
(b) The scale 1 cm represents 25 m can be written in the form 1 : k. Find the value of k. [1 mark]
(ii) k =
(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 & Answer
(a)
240 ÷ 25 = 9.6 cm B1
Answer (a): 9.6 cm cm
(b)
25 m = 2500 cm, so 1 : 2500 ⇒ k = 2500 B1
Answer (b): (ii) k = k = 2500
(b)
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
Question Paper (PDF) • Mark Scheme (PDF)
Score: 0 / 10 (0 / 35 marks)
Loci and construction Worksheets Next: Combining ratios →

Grade 4 Quizzes

24 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