Mathematics · Lesson 16

Loci & constructions

JAMBWAECNECO

What the syllabus asks

JAMB: construction and locus

WAEC: construction with ruler and compasses only: bisecting a line segment and an angle; perpendiculars to a line from and at a point; angles of 90°, 60°, 45°, 30°, 120°, 135°, 75°, 105° and their combinations; parallel lines; triangles and quadrilaterals from given data; dividing a line segment in a given ratio; loci: in two dimensions (fixed distance from a point or a line, equidistant from two…

NECO: construction with ruler and compasses only: bisecting a line segment and an angle; perpendiculars to a line from and at a point; angles of 90°, 60°, 45°, 30°, 120°, 135°, 75°, 105° and their combinations; parallel lines; triangles and quadrilaterals from given data; dividing a line segment in a given ratio; loci: in two dimensions (fixed distance from a point or a line, equidistant from two…

What you will learn

Define a locus and describe the four basic loci in two dimensions.
Bisect a line segment and an angle with ruler and compasses.
Construct perpendiculars from and at a point on a line, and a line parallel to a given line.
Construct angles of 60°, 30°, 90°, 45°, 120°, 135°, 75° and 105°.
Construct triangles and quadrilaterals from given data and measure unknown sides and angles.
Divide a line segment in a given ratio.
Solve problems using intersecting loci and write simple loci as equations.

Key points

A locus is the set of points satisfying a condition; describe it by name and details.
Fixed distance from a point: circle; from a line: two parallel lines.
Equidistant from two points: perpendicular bisector; from two lines: angle bisectors.
Use only ruler and compasses; leave all arcs visible.
Perpendicular bisector: radius more than half the line, equal arcs from both ends.
Angle bisector: arc from the vertex, then equal arcs from the two cut points.
60° from equilateral arcs; 30°, 15°, 45° by bisecting; 90° perpendicular; 75° and 105° by bisecting between.
Triangles from SSS, SAS, ASA or RHS; check with sine/cosine rules; triangle inequality.
Divide a line in a ratio using equal steps on a ray and parallel lines.
Intersections of loci give required points; inequalities give regions; coordinate loci by PA² = PB² and x² + y² = r².

Start here: What a locus is

Tie a goat to a peg with a 5 m rope and let it walk with the rope tight: it traces a circle. The path of a point that moves according to a rule is called a locus.

A locus is the set of all points that satisfy a given condition, or the path traced by a point moving according to that condition.

To find a locus, ask: "Where could the point be?" Mark several possible positions, see the pattern, then describe it in words and draw it accurately.

Everyday examples:

The tip of a clock’s minute hand: a circle.
The centre (hub) of a bicycle wheel rolling along a straight, level road: a straight line parallel to the road.
A boy walking so that he is always the same distance from two goalposts: the perpendicular bisector of the line joining them.
A swing seat: an arc of a circle.

Worked example 16.1

Describe the locus of (a) a point 3 cm from a fixed point O, (b) the centre of a coin of radius 1 cm rolling around the outside of a fixed plate of radius 6 cm.

(a) A circle with centre O and radius 3 cm.
(b) The centre of the coin is always 6 + 1 = 7 cm from the plate’s centre: a circle of radius 7 cm, concentric with the plate.

Circle radius 3 cm; circle radius 7 cm

Loci are not just a classroom exercise. Surveyors use them to fix boundaries; engineers use them to position telecommunication masts so that every village is within range; the Nigerian Air Force and air-traffic controllers use the intersection of distance circles from radar stations to locate aircraft; and your phone’s GPS finds your position as the intersection of spheres centred on several satellites. In every case the idea is the same: each measurement gives a locus, and the answer is where the loci meet.

💡 Exam tip: a locus answer has two parts: a name (circle, line, bisector) and its details (centre and radius; which points; which lines).

Try it yourself. Describe the locus of a point that is always 2 m above a flat floor (in a room).

Show solution

A plane parallel to the floor, 2 m above it. (In 2D questions, a line parallel to the floor line.)

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Practice questions

1. Describe the locus of points 2 cm from a straight line segment 6 cm long.

Show worked answer

Two parallel lines 2 cm either side, joined by semicircles of radius 2 cm at the ends.

2. List the steps to construct an angle of 45°.

Show worked answer

Construct a perpendicular (90°) at a point, then bisect it.

3. Construct triangle ABC with AB = 6 cm, BC = 5 cm and angle ABC = 90°. Calculate AC.

Show worked answer

√(36 + 25) = √61 ≈ 7.8 cm.

4. How would you construct 135°?

Show worked answer

Construct 90°, then bisect the 90° between the perpendicular and the extended line, giving 90° + 45°.

5. Divide a 12 cm line in the ratio 1 : 3. Find the parts.

Show worked answer

3 cm and 9 cm.

Key words

Locus: The set of all points that satisfy a given condition; plural loci.
Construction: An accurate drawing made with ruler and compasses only.
Arc: Part of the circumference of a circle drawn with compasses.
Bisect: Divide into two equal parts.
Perpendicular bisector: The line at right angles to a segment through its midpoint.
Angle bisector: The line dividing an angle into two equal angles.
Perpendicular: At right angles (90°) to a line.
Equidistant: At equal distances.
Radius: The fixed distance set on the compasses.
Intersection: The point(s) where two lines or curves meet.
Parallel lines: Lines that remain the same distance apart and never meet.
Equilateral triangle: A triangle with all sides equal and all angles 60°.

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