Secondary 4 constructions and loci become easier when students translate each verbal condition into a geometric object. This Mathematics tuition guide for Punggol families explains perpendicular bisectors, angle bisectors, equidistant points and locus regions through original worked examples.
A student may know how to draw a perpendicular bisector with a compass but not recognise that “equidistant from A and B” is asking for that same locus. Another may shade the wrong side of a boundary because the condition is not tested.
At eduKatePunggol, Secondary 4 Mathematics is taught in focused groups of up to three students in 1.5-hour lessons near Punggol MRT. This guide supports the wider Secondary 4 Mathematics year plan. Use the constructions appropriate to the student’s current syllabus and school programme.
A locus is a set of points satisfying a condition
Instead of asking for one point, a locus asks for every point that satisfies a geometric rule.
For example, every point exactly 3 cm from a fixed point P lies on a circle of radius 3 cm centred at P.
The words define the geometry.
Equidistant from two points means perpendicular bisector
The set of all points equally distant from A and B lies on the perpendicular bisector of AB.
This line:
- passes through the midpoint of AB;
- meets AB at 90°;
- contains all points P for which PA = PB.
Worked example 1: construct a perpendicular bisector
Given segment AB, draw arcs from A and B using the same compass radius greater than half AB. The arcs intersect above and below the segment.
Join the two arc intersections. The resulting line is the perpendicular bisector of AB.
Do not erase the construction arcs if the question requires a construction. They show how the line was created.
Equidistant from two lines means angle bisector
For two intersecting lines, points equally distant from both lines lie on angle bisectors.
This is why “equidistant from sides AB and AC” inside angle BAC points toward the angle bisector of ∠BAC.
Worked example 2: construct an angle bisector
From the angle vertex, draw an arc cutting both arms. From those two cut points, draw equal-radius arcs that intersect inside the angle.
Join the vertex to the new intersection. This line bisects the angle.
The construction works because the new point is built symmetrically from the two angle arms.
A fixed distance from a line creates parallel loci
All points exactly 2 cm from a straight line lie on two lines parallel to the original, one on each side, each 2 cm away.
If a diagram restricts the permitted region—for example inside a rectangle—only the relevant portions of those parallel loci are used.
Worked example 3: combine two locus conditions
Find points that are:
- equidistant from A and B; and
- exactly 4 cm from point C.
The first locus is the perpendicular bisector of AB.
The second locus is a circle centred at C with radius 4 cm.
The required points are where the two loci intersect.
The answer is therefore found by combining conditions, not by guessing a convenient point.
Inequalities in loci require region testing
Suppose a condition says points are closer to A than to B.
The perpendicular bisector is the boundary where distances are equal. The solution region lies on the A side of that boundary.
A simple test point near A confirms which side to shade.
This connects to the inequalities guide: a boundary separates permitted and non-permitted regions.
Construction accuracy matters
Use a sharp pencil, sensible compass width and visible arcs. Tiny arcs that barely intersect are hard to inspect. Very thick lines make exact intersections ambiguous.
Construction questions assess geometric method as well as the final line.
How we diagnose constructions and loci mistakes
Translation error: the verbal condition is matched to the wrong locus.
Construction error: unequal compass radii or poorly placed arcs destroy symmetry.
Boundary error: the correct locus is drawn but the wrong region is selected.
Combination error: two separate conditions are found but their intersection is not identified.
Presentation error: construction arcs are removed or the final region is not labelled clearly.
Why the three-student format helps
In a group of up to three students, one learner can translate the condition, another can perform the construction and another can test the final region. The tutor can correct the first wrong interpretation before the whole diagram is built around it.
What a 90-minute lesson could look like
An illustrative lesson could use ten minutes for condition-to-locus matching, twenty minutes on perpendicular bisectors, twenty minutes on angle bisectors, twenty minutes on combined loci and regions, and twenty minutes for independent constructions, error review and continuation work.
Repair, stabilisation and extension
Repair: use one condition at a time and name the locus before constructing it.
Stabilisation: mix point-distance, line-distance and equidistance conditions.
Extension: combine two or three conditions and require the student to identify the final feasible region precisely.
Try a short independent check
- What is the locus of points 5 cm from point P?
- What is the locus of points equidistant from A and B?
- What is the locus of points equidistant from two intersecting lines?
Answers: a circle centred at P with radius 5 cm; the perpendicular bisector of AB; and the angle bisectors of the two lines.
What progress should look like
- verbal conditions are translated into the correct locus;
- construction arcs are symmetric and visible;
- perpendicular and angle bisectors are distinguished;
- region inequalities are tested correctly;
- multiple conditions are combined by intersection;
- the final construction remains readable.
Punggol class details and consultation inputs
eduKatePunggol Secondary Mathematics tutorials run for 1.5 hours in groups of up to three students near Punggol MRT. Confirm current availability, fees and meeting arrangements directly.
Bring the student’s subject level, examination year, compass and recent construction questions. Original arcs should remain visible where possible because they show whether the difficulty was interpretation or execution.
Frequently asked questions
Why is a perpendicular bisector a locus?
Every point on it is equally distant from the two endpoints of the segment.
How do I know which side of a locus boundary to shade?
Choose a simple test point and see whether it satisfies the stated condition.
Turn each condition into a geometric object
Return to the Secondary 4 Mathematics year plan for the wider revision runway. For related spatial transformations, use the transformations guide.
Name the condition, construct the locus and test the final region. Families can WhatsApp eduKatePunggol with recent work to discuss a suitable next step.
