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Sec 4 E Maths: Constructions and Loci, practice questions & worked solutions

Constructions and loci questions selected from 2023 to 2025 Singapore school E Maths prelim papers, each with the school’s full worked solution.

6 questionsStep-by-step solutionsFree to access

About this topic & key methods

Constructions in E Maths use a ruler and compasses to draw triangles, perpendicular bisectors and angle bisectors, then use them to find points or regions that meet given conditions. All six questions here are Paper 1 items, and each solution shows the finished construction with its arcs.

Attempt each question on paper before opening its worked solution. The solutions are the schools’ own marking schemes, so some steps are brief.

Key methods for constructions and loci

Triangle from three sides: draw one side, then arcs of the other two lengths from its ends; they meet at the third vertex
Points equidistant from $A$ and $B$ lie on the perpendicular bisector of $AB$
Points equidistant from two lines lie on the bisector of the angle between them
The perpendicular bisectors of two chords meet at the centre of the circle
Nearer to $A$ than to $B$ is the side of the perpendicular bisector containing $A$ ; nearer to one line is the side of the angle bisector containing that line
With a scale of $1$ cm to $k$ km, divide real distances by $k$ to draw, and multiply measured lengths by $k$
Leave all construction arcs visible

For lessons that build these methods week by week, see our E Maths tuition for Sec 3 and Sec 4.

Constructions and Loci questions with worked solutions

Unless the question specifies otherwise, give non-exact answers to 3 significant figures and angles in degrees to 1 decimal place.

Q1Tampines Secondary School2024Prelim P1 Q7●○○ Foundational2 marks

Circle through three points using perpendicular bisectors, Tampines Secondary School, 2024 Prelim P1 Q7

Use a ruler and compasses only for this question.
You must show all your construction arcs.

Line $L$ is the perpendicular bisector of $FG$ .
Construct a circle that passes through the points $F$ , $G$ and $H$ . [2]

Tampines Secondary School 2024 P1 Q7: diagram for the question

Show worked solution▾

Tampines Secondary School 2024 P1 Q7: worked solution diagram

Answer: Construct the perpendicular bisector of $GH$ . It meets $L$ at the centre $O$ . Draw the circle with centre $O$ and radius $OG$ ; it passes through $F$ , $G$ and $H$ .

Q2Fairfield Methodist School (Secondary)2023Prelim P1 Q16●○○ Foundational3 marks

Perpendicular bisector and angle bisector to locate a region, Fairfield Methodist School (Secondary), 2023 Prelim P1 Q16

The diagram below represents a plot of land, $PQRS$ , which is to be used for a park.

Fairfield Methodist School (Secondary) 2023 P1 Q16: diagram for the question

Construct the perpendicular bisector of $PQ$ . [1]
Construct the bisector of angle $PSR$ . [1]
A children's playground is to be built in the park. The planned location of the playground is nearer to $Q$ than to $P$ , and nearer to $PS$ than to $RS$ .
Shade the region where the playground can be built. [1]

Show worked solution▾

(a), (b), (c)

Fairfield Methodist School (Secondary) 2023 P1 Q16: worked solution diagram

Answer: (c) the region inside $PQRS$ on $Q$ 's side of the perpendicular bisector of $PQ$ and on $PS$ 's side of the bisector of angle $PSR$

Q3Anglo-Chinese School (Barker Road)2023Prelim P1 Q9●○○ Foundational3 marks

Angle bisector and perpendicular bisector to shade a region, Anglo-Chinese School (Barker Road), 2023 Prelim P1 Q9

The diagram shows a triangle $ABC$ .

Anglo-Chinese School (Barker Road) 2023 P1 Q9: diagram for the question

Construct the angle bisector of angle $ACB$ . [1]
Construct the perpendicular bisector of $BC$ . [1]
Shade the region closer to $AC$ than to $BC$ and closer to $C$ than to $B$ . [1]

Show worked solution▾

(a)

Evidence of construction marks for angle bisector at angle $C$ .

(b)

Evidence of construction marks for perpendicular bisector of $BC$ .

(c)

Anglo-Chinese School (Barker Road) 2023 P1 Q9: worked solution diagram

Answer: (c) the region between $AC$ and the bisector of angle $ACB$ , on the $C$ side of the perpendicular bisector of $BC$ (shaded in the diagram)

Q4Cedar Girls’ Secondary School2025Prelim P1 Q20●●○ Standard3 marks

Constructing a triangle from its sides and finding the circumcentre, Cedar Girls’ Secondary School, 2025 Prelim P1 Q20

There are 3 statues located at points $A$ , $B$ and $C$ respectively.
The diagram in the space below shows the positions of points $A$ and $B$ .

It is given that $AC$ is 8 km and $BC$ is 10 km.
Using appropriate constructions, find and label the position of point $C$ . [1]
There is a treasure chest buried in a location that is equidistant from $A$ , $B$ and $C$ . Find and label this point $X$ in the diagram above. [2]

Show worked solution▾

(a)

Draw 2 arcs

Find $C$ , the intersection of the 2 arcs.

Cedar Girls’ Secondary School 2025 P1 Q20: worked solution diagram

(b)

Draw at least 2 perpendicular bisectors

Find $X$ , the intersection of the 2 perpendicular bisectors

Answer: (a) $C$ is where the arc of radius 8 cm centred at $A$ meets the arc of radius 10 cm centred at $B$ (b) $X$ is where the perpendicular bisectors of the sides of triangle $ABC$ meet

Q5School of Science and Technology2025Prelim P1 Q23●●○ Standard4 marks

Centre of an arc from its tangents and a region defined by two loci, School of Science and Technology, 2025 Prelim P1 Q23

The diagram shows a plot of land, $PTVU$ , on a campsite.
$PT$ and $PU$ are tangents of an arc of a circle at $T$ and $U$ .

School of Science and Technology 2025 P1 Q23: diagram for the question

Find the centre of the circle and label it $X$ . [1]
Construct the perpendicular bisector of $TP$ . [1]
Construct the angle bisector of angle $TPU$ . [1]
A lookout tower is to be built on the campsite $PTVU$ , nearer to $T$ than $P$ and nearer to $PT$ than $PU$ .
Shade the region where the lookout tower is to be built. [1]

Show worked solution▾

School of Science and Technology 2025 P1 Q23: worked solution diagram

(a)

Find the centre of the circle and label it $X$ .

Construction of perpendicular lines at $T$ and $U$
Labelling of intersection of perpendicular lines at $T$ and $U$ as point $X$

(b)

Construction of the perpendicular bisector of $TP$ with working

(c)

Construction of the angle bisector of angle $TPU$ with working

(d)

A lookout tower is to be built on the campsite $PTVU$ , nearer to $T$ than $P$ and nearer to $PT$ than $PU$ .
Shading of the region where the lookout tower is to be built.

Answer: (a) $X$ is where the perpendicular to $PT$ at $T$ meets the perpendicular to $PU$ at $U$ (d) the region inside $PTVU$ on $T$ 's side of the perpendicular bisector of $TP$ and on $PT$ 's side of the bisector of angle $TPU$

Q6Nan Chiau High School2024Prelim P1 Q24●●● Challenging5 marks

Scale drawing with two loci to fix a point, Nan Chiau High School, 2024 Prelim P1 Q24

$ABCD$ represents a plot of flat land, drawn on a scale of 1 cm : 15 km.

Nan Chiau High School 2024 P1 Q24: diagram for the question

Construct and mark clearly point $B$ such that its actual distance is 145.5 km from point $A$ and 109.5 km from point $C$ . [1]
A vertical Tower, $T$ , is equidistant from $A$ and $C$ and equidistant from $AD$ and $CD$ .
With clear constructions, mark the letter, $T$ on the diagram to indicate the position of the Tower. [2]
Hence, measure and calculate the actual distance from $T$ to $C$ . [2]

Show worked solution▾

(a) and (b)

Nan Chiau High School 2024 P1 Q24: worked solution diagram

(c)

$5.9\times15\ =88.5$ km

Answer: (c) $88.5$ km

Need help with constructions and loci? Our Sec 4 E Maths tuition classes work through prelim questions like these every week.

More practice related to constructions and loci

Sec 4 E Maths: Angles and Polygons

Sec 4 E Maths: Congruence and Similarity

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