Secondary 2 constructions and loci become clearer when students stop seeing compass-and-ruler work as a sequence of mysterious arcs. Every construction creates a geometric condition, and every locus describes all points satisfying a condition.
At eduKate Punggol, our Secondary 2 Mathematics tutorials have up to three students and normally run for around 90 minutes. That gives the tutor room to inspect whether a learner understands what the construction means, not merely whether the final arc looks neat.
This guide uses original examples for students whose current school programme includes construction or locus reasoning. It supports the wider Punggol Secondary 2 Mathematics Tutor guide and our broader work on angle properties and geometric reasoning.
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What Is a Locus?
A locus is the set of all points satisfying a stated condition.
For example, all points exactly 4 cm from a fixed point P lie on a circle centred at P with radius 4 cm.
The locus is not one chosen point. It is the entire set of possible points that satisfy the rule.
Locus 1: Fixed Distance From One Point
If a point must stay 5 cm from A, draw a circle centred at A with radius 5 cm.
Every point on the circle is 5 cm from A. Points inside or outside the circle fail the exact-distance condition.
Locus 2: Equal Distance From Two Points
All points equidistant from A and B lie on the perpendicular bisector of AB.
This is not a memorised coincidence. Any point on the perpendicular bisector forms equal distances to A and B.
Worked Example 1: Construct the Perpendicular Bisector
Given a segment AB, set the compass to a radius greater than half of AB.
- Draw an arc from A above and below the segment.
- Without changing the compass width, draw matching arcs from B.
- Join the two arc-intersection points.
- The resulting line is the perpendicular bisector of AB.
The construction creates points that are equally distant from A and B. Joining them gives the full locus.
Locus 3: Equal Distance From Two Intersecting Lines
The points equidistant from two intersecting lines lie on their angle bisectors.
Both the internal and external angle bisectors satisfy the equal-distance condition.
Students often remember only the internal bisector because it is visually more obvious.
Worked Example 2: Construct an Angle Bisector
Given angle ABC:
- Draw an arc centred at B that crosses BA and BC.
- From the two crossing points, draw equal-radius arcs that meet inside the angle.
- Join B to the intersection of those arcs.
The new ray bisects the angle. More importantly, every point on it is equidistant from the two sides of the angle.
Locus 4: Fixed Distance From a Line
All points exactly 3 cm from a straight line lie on two parallel lines, each 3 cm from the original line.
There are two sides to the original line, so the complete locus contains two parallel components.
Worked Example 3: Combine Two Loci
A point X must be 4 cm from A and equidistant from A and B.
The first condition gives a circle centred at A with radius 4 cm.
The second condition gives the perpendicular bisector of AB.
Possible positions of X are the intersection points of those two loci, if such intersections exist.
This is the key idea in combined locus problems: draw each condition separately, then take the overlap.
Region Questions Need Boundaries and Inequalities
If the condition changes from “exactly 4 cm from A” to “less than 4 cm from A”, the answer is the interior of the circle, not merely the circumference.
If the condition says “at least 4 cm from A”, the valid region is on or outside the circle, depending on the exact wording and diagram boundary.
Words such as exactly, less than, more than, nearer to and equidistant have geometric consequences.
Worked Example 4: Nearest to A Rather Than B
Given points A and B, the perpendicular bisector separates the plane into two half-planes.
Points on A’s side of the perpendicular bisector are nearer to A than to B. Points on the other side are nearer to B. Points on the bisector are equidistant.
This turns a construction into a decision boundary.
Five Common Construction-and-Locus Errors
- Using a compass width that changes midway through a construction.
- Drawing only part of the locus instead of the complete set.
- Forgetting the external angle bisector when the condition includes all points equidistant from two intersecting lines.
- Confusing an exact-distance locus with a within-distance region.
- Drawing two correct loci but failing to take their intersection as the final answer.
Why a 3-Pax Tutorial Helps
One student may reproduce the arc sequence without knowing what it proves. Another may understand the locus but construct it inaccurately. A third may draw both conditions correctly but choose the wrong overlap region.
In a class of three, the tutor can ask the learner to state the condition before drawing. That makes the geometry visible before the instrument work begins.
An Illustrative 90-Minute Lesson
- Review perpendicular and equal-distance ideas.
- Construct one perpendicular bisector.
- Connect it explicitly to the equal-distance locus.
- Construct an angle bisector.
- Compare exact distance with within-distance regions.
- Combine two loci and identify intersections.
- Finish with an independent word-to-diagram problem.
Try Four Questions
- Describe the locus of points exactly 6 cm from P.
- Describe the locus of points equidistant from A and B.
- Describe the locus of points exactly 2 cm from a straight line l.
- A point must be exactly 5 cm from A and equidistant from A and B. What construction identifies its possible positions?
Answers: (1) A circle centred at P, radius 6 cm. (2) The perpendicular bisector of AB. (3) Two lines parallel to l, each 2 cm away. (4) Intersections of the circle centred at A radius 5 cm with the perpendicular bisector of AB.
What Progress Should Look Like
- The student states the locus condition before drawing.
- Compass widths are preserved deliberately.
- Perpendicular and angle bisectors are connected to equal-distance meaning.
- Exact-distance loci are distinguished from shaded regions.
- Combined conditions are solved by intersecting loci.
- The learner can explain why the construction works.
Full Subject-Based Banding and School Scope
Students may take Mathematics at G1, G2 or G3 subject levels, and schools may sequence construction or locus content differently. Use the student’s actual school programme to decide which constructions are current and which are extension.
The durable habit is to translate words into geometric conditions before using the compass and ruler.
Class Details
Format: Premium 3-pax small-group tutorials
Level: Secondary 2 Mathematics
Subject support: matched to the student’s current Mathematics subject level and school programme
Duration: normally around 90 minutes weekly
Location: 83 Punggol Central, Singapore 828761
Teaching approach: first-principles explanation, diagram reasoning, guided and independent practice, retrieval, error analysis, school-paper alignment and carefully paced extension.
What Parents Can Bring to the Consultation
- recent construction worksheets;
- a compass and ruler used by the student;
- marked geometry papers;
- questions involving combined loci or shaded regions; and
- the school’s current topic sequence.
Frequently Asked Questions
Why is the perpendicular bisector a locus?
Because every point on it is equidistant from the two endpoints, and every point equidistant from the endpoints lies on that bisector.
Why are there two lines for a fixed distance from a line?
The condition can be satisfied on either side of the original line.
What is the difference between a locus and a region?
A locus may be a boundary such as a circle. A region condition such as “within 4 cm” includes an area, not only the boundary.
When is tuition useful?
When the student can imitate constructions but cannot translate the written condition or explain the resulting locus independently.
Helpful Reading and Next Step
Continue with angle properties and parallel lines, transformations, and coordinate geometry.
The objective is a student who can convert a condition into a geometric object and justify why every valid point belongs there. Discuss your child’s current Mathematics work with eduKate Punggol.
