SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 16
Construction turns geometric conditions into exact spatial objects. Scale drawings translate one set of lengths into another while preserving proportion. Loci describe every point satisfying a condition. Together, these ideas make geometry operational: instead of merely calculating from a finished diagram, the learner builds or identifies the diagram from rules.
A perpendicular bisector is not just a line drawn through the middle of a segment. It is the locus of points equidistant from the segment’s endpoints. An angle bisector is not simply “half an angle” by appearance; it is the locus of points equidistant from the two sides of the angle. A scale drawing does not preserve actual lengths, but it preserves their ratios.
This guide develops ruler-and-compass reasoning, perpendicular and angle bisectors, simple bearings, scale drawings and elementary loci. Exact course sequencing varies across subject levels and schools, so use the MOE secondary syllabus directory as the official reference.
Return to the Secondary Mathematics Hub. Revisit Geometry, Angles and Polygons for the properties used here and Ratio and Proportion for scale-factor reasoning.
Navigate: construction meaning · perpendicular bisector · angle bisector · scale drawings · bearings · loci · combined loci · practice · answers.
1. Construction is controlled geometry, not freehand drawing
A geometric construction uses stated tools and properties to create an object satisfying exact conditions. In traditional school construction, an unmarked straightedge and compass are central tools, though school tasks may also permit rulers, protractors and set squares depending on the exercise.
The purpose is not artistic neatness. The purpose is to encode a condition accurately enough that the resulting object has the required property.
Construction versus measurement
Measuring an angle estimates the value from a drawing. Constructing an angle bisector creates a line whose defining property guarantees equal angular division under the construction.
2. Constructing a perpendicular through a point on a line
Suppose point P lies on line l and we want a line through P perpendicular to l.
One classical construction is:
- With centre P, draw an arc cutting l at A and B.
- With equal radius greater than half AB, draw arcs centred at A and B that intersect at C.
- Join P to C.
PC is perpendicular to AB.
Why it works
PA = PB by the first circle. CA = CB by the equal-radius arcs. Triangles PAC and PBC are congruent, forcing equal adjacent angles at P. Since they form a straight line, each is 90°.
3. A perpendicular bisector combines midpoint and right angle
The perpendicular bisector of segment AB passes through the midpoint of AB and meets AB at 90°.
Classical construction
- Set a compass radius greater than half AB.
- Draw arcs centred at A above and below the segment.
- Without changing the radius, draw arcs centred at B to intersect the first pair.
- Join the two arc intersections.
The joining line is the perpendicular bisector.
Core locus property
Every point on the perpendicular bisector of AB is equidistant from A and B. Conversely, every point equidistant from A and B lies on the perpendicular bisector in the plane.
4. Use perpendicular-bisector reasoning without constructing every time
If point P is known to lie on the perpendicular bisector of AB and PA = 7 cm, then PB = 7 cm.
The result follows from the locus property. You do not need to re-prove the whole construction unless the question asks for justification.
Reverse reasoning
If PA = PB, then P lies on the perpendicular bisector of AB. This is useful when classifying a point from distance conditions.
5. An angle bisector divides an angle into two equal angles
To construct the bisector of angle ABC:
- With centre B, draw an arc crossing BA and BC at D and E.
- With equal radius, draw arcs centred at D and E that intersect at F inside the angle.
- Join B to F.
BF bisects angle ABC.
Why it works
BD = BE from the first arc. DF = EF from equal-radius arcs. BF is common. The two triangles are congruent, so the two angles at B are equal.
6. The angle bisector is also a locus
Points on the internal angle bisector are equidistant from the two sides of the angle, where distance from a point to a line means perpendicular distance.
Worked interpretation
A point P lies inside an angle and is 4 cm from one side. If P lies on the angle bisector, its perpendicular distance from the other side is also 4 cm.
Do not measure along a sloping path to the side. Point-to-line distance is perpendicular.
7. A scale drawing preserves ratios of corresponding lengths
A scale of 1:500 means 1 unit on the drawing represents 500 of the same units in reality.
Worked example
At scale 1:500, a drawn length of 6 cm represents 6 × 500 = 3000 cm = 30 m.
The units must be made compatible before interpreting the final distance.
8. Reverse a scale by division
A real distance of 45 m is to be represented at scale 1:750. Convert 45 m to 4500 cm. Drawing length = 4500 ÷ 750 = 6 cm.
Direction check
At a reduction scale such as 1:750, the drawing should be much smaller than the real object. An answer of 600 cm would fail the scale check.
9. Scale factor affects area differently
If corresponding lengths scale by k, corresponding areas of similar figures scale by k².
At a drawing scale 1:100, a 2 cm by 3 cm rectangle represents a 200 cm by 300 cm rectangle. The drawing area is 6 cm² while the real area is 60,000 cm², a factor of 10,000 = 100².
Treat formal area-scale work as an extension if it has not yet appeared in your course.
10. Bearings describe direction clockwise from north
A three-figure bearing is measured clockwise from north and written using three digits. East is 090°, south 180°, west 270° and north 000° or 360° depending on context.
Worked example
A point lies on a bearing of 060° from A. Start from north at A and rotate clockwise 60°.
Do not start measuring from the east-facing horizontal axis as in ordinary coordinate-angle work.
11. Reverse bearings differ by 180°
If the bearing of B from A is 070°, then the bearing of A from B is 070° + 180° = 250°.
If the original bearing exceeds 180°, subtract 180° to obtain the reverse bearing.
Worked example
Bearing of Q from P is 235°. Bearing of P from Q is 235 − 180 = 055°.
12. Scale drawings can combine distance and bearing
Suppose B is 8 km from A on bearing 120°. At a map scale where 1 cm represents 2 km, draw a north line at A, measure 120° clockwise, and mark B 4 cm along the ray.
The construction combines an angular condition and a length scale.
Reading back
If the finished drawing is accurate, it can support approximate measurement of another distance or bearing. Label such measured results as approximate unless exact relationships justify them.
13. A locus is the set of all points satisfying a condition
The word “locus” means a set of positions defined by a rule.
Common simple loci
| Condition | Locus |
|---|---|
| Exactly r units from a fixed point A | Circle centred at A with radius r |
| Equidistant from fixed points A and B | Perpendicular bisector of AB |
| Exactly d units from a straight line | Two lines parallel to the original, distance d away, subject to region boundaries |
| Equidistant from two intersecting lines | The angle bisectors |
14. “Within” and “at least” produce regions rather than single curves
Points exactly 5 cm from A lie on a circle. Points within 5 cm of A lie inside or on that circle.
Points more than 5 cm from A lie outside the circle. Boundary inclusion depends on words such as “at least”, “more than” and “no more than”.
Connection to inequalities
Geometric regions often behave like inequalities in space. A boundary may be included or excluded, and several conditions may intersect.
15. Combined loci require every condition to be satisfied
Suppose a point P must be equidistant from A and B and exactly 4 cm from A.
The first condition gives the perpendicular bisector of AB. The second gives a circle centred at A with radius 4 cm. Valid positions are the intersection points of those two loci.
No intersection means no solution
If the two loci do not meet, no point satisfies both conditions simultaneously.
16. Regions can be narrowed by multiple constraints
A point must be within 6 cm of A and closer to B than to C. The first condition gives the disk of radius 6 around A. The second condition selects the half-plane on B’s side of the perpendicular bisector of BC.
The valid region is their overlap.
Test-point method
When deciding which side of a boundary to shade, choose a simple test point and check the condition directly.
17. Construction accuracy depends on the preserved condition
If compass radius changes unintentionally during an equal-radius construction, the theoretical congruence argument may no longer apply.
If a scale drawing uses the wrong conversion factor, every later measured conclusion inherits the error.
Working marks matter
When construction arcs are required, leave them visible unless told otherwise. They show how the line was generated and provide evidence that the construction method was used.
18. Common construction and loci errors
| Error | Likely issue | Repair prompt |
|---|---|---|
| Perpendicular bisector drawn through midpoint but not at 90° | Only one defining property used | Which two conditions must both hold? |
| Scale 1:500 interpreted as 1 cm = 500 m | Units ignored | What same units does the ratio compare? |
| Bearing measured anticlockwise | Bearing convention confused | Where do bearings start and turn? |
| “Within 4 cm” drawn as only the circle | Boundary confused with region | Do interior points also satisfy the condition? |
| Uses midpoint line for points equidistant from A and B without perpendicularity | Locus property incomplete | What is the full locus of equal distance? |
19. Practice laboratory
- State the two defining properties of a perpendicular bisector.
- What locus contains all points equidistant from A and B?
- What locus contains all points exactly 3 cm from A?
- What region contains all points no more than 3 cm from A?
- At scale 1:400, what real distance does 7 cm represent?
- At scale 1:2500, how long on the drawing represents 50 m?
- Find the reverse bearing of 035°.
- Find the reverse bearing of 240°.
- A point is on the angle bisector and 5 cm from one side of the angle. Find its perpendicular distance from the other side.
- A point must be equidistant from A and B and 6 cm from A. Describe how to locate all possible positions.
- Explain why working arcs are useful evidence in a construction.
- A map uses 1 cm to represent 4 km. Two locations are 6.5 cm apart. Find their real distance.
- Why is “closer to A than B” represented by one side of the perpendicular bisector of AB?
- A point must be within 5 cm of P and at least 2 cm from line l. Describe the required region.
20. Explained answers
1. It passes through the midpoint of the segment and is perpendicular to the segment.
2. The perpendicular bisector of AB.
3. A circle centred at A with radius 3 cm.
4. The disk inside and including that circle.
5. 7 × 400 = 2800 cm = 28 m.
6. 50 m = 5000 cm. 5000 ÷ 2500 = 2 cm.
7. 035° + 180° = 215°.
8. 240° − 180° = 060°.
9. 5 cm.
10. Construct the perpendicular bisector of AB and a circle centred at A with radius 6 cm. Their intersection points are the possible positions.
11. They show the equal-radius relationships used to generate the constructed line.
12. 6.5 × 4 = 26 km.
13. The perpendicular bisector is the equal-distance boundary. Every point on A’s side is closer to A; every point on B’s side is closer to B.
14. Intersect the disk within 5 cm of P with the region outside the two parallel boundary lines that lie 2 cm from l, including the 2 cm boundaries if “at least” is inclusive.
21. Complete mixed problem
Problem: Town B is 12 km from Town A on bearing 060°. Town C must be equidistant from A and B and within 8 km of A. A map uses 1 cm to represent 2 km. Describe a construction that shows all possible map positions of C.
First draw north at A. Since 1 cm represents 2 km, AB = 6 cm on the map. Measure 060° clockwise from north and mark B 6 cm from A.
Construct the perpendicular bisector of AB. This is the locus of points equidistant from A and B.
Within 8 km of A corresponds to within 4 cm on the map. Draw a circle centred at A with radius 4 cm and consider its interior.
The possible positions of C are the parts of the perpendicular bisector lying inside or on that circle.
22. Teaching construction through properties
After performing a construction, ask the learner to name the property that was created. “This line is perpendicular” is only part of a perpendicular-bisector answer; “and it passes through the midpoint” completes it.
Then reverse the direction: give the property and ask which construction or locus it defines.
Combine only after the single loci are secure
Students should first recognise circles, perpendicular bisectors and angle bisectors independently. Combined-region questions then become an intersection task rather than a new mystery topic.
23. Questions students often ask
Do I need a protractor for every construction?
No. Classical perpendicular and angle-bisector constructions can be made with compass and straightedge. Follow the tools allowed by the task.
Why are bearings three figures?
The three-digit format removes ambiguity: 5° is written 005°, while 50° is 050°.
Does a scale drawing give exact real measurements?
The scale relationship is exact in the mathematical model, but measurements read from a hand-drawn diagram may carry construction and ruler error.
What is the difference between a locus and a region?
A locus can be a line or curve representing exact equality conditions. Inequality conditions such as “within” often produce a two-dimensional region.
What is the best check?
Test whether points on the constructed object actually satisfy the defining condition.
24. Return path
Construction, scale and loci connect geometry with proportion and inequalities. Revisit Geometry, Angles and Polygons for shape properties, Ratio and Proportion for scale, and Linear Inequalities and Number-Line Reasoning for boundary inclusion and solution regions.
Sources and learning boundaries
Official curriculum reference: MOE Secondary Syllabus Directory. Construction, scale drawing, bearings and locus depth vary across subject levels and school sequencing.
The town, map and distance examples are independently constructed teaching examples and not navigation instructions for actual locations.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Translate each spatial condition into a construction or locus, preserve scale and orientation, intersect constraints and verify the resulting region.
