Grade 9 Loci and Constructions: Short Notes

9 ශ්‍රේණිය · 2 වාරය · 13 පරිච්ඡේදය · පථ හා නිර්මාණ

68%

of Grade 9 term tests include it (27 of 40)

314

marks in past papers (43 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

Name and draw the four basic loci
Locate a point that satisfies two conditions by intersecting loci
Construct the perpendicular bisector of a line segment and the bisector of an angle
Construct 60°, 90°, 30° and 45° angles with compasses and straight edge

Where can the goat graze?

A goat on a farm near Anuradhapura is tied to a peg with a 4 m rope. Where can it reach? Every point up to 4 m from the peg, a full circle of grass. Describing all the points that obey a rule like this is what loci are about.

What is a locus?

A locus is the set of all points that satisfy a given condition. (Plural: loci.)Think of it as the path traced by a point that moves while always obeying the rule.There are four basic loci you must know by heart. Each one can be drawn with a ruler and a pair of compasses.

Loci 1 and 2

1. Fixed distance r from a point O: a circle, centre O, radius r.2. Equidistant from two points A and B: the perpendicular bisector of AB. Any P on it has PA = PB.

Loci 3 and 4

3. Fixed distance d from a line l: two lines parallel to l, one on each side, at distance d.4. Equidistant from two intersecting lines: the bisectors of the angles between them. P is the same distance from both lines.

⚠️ Don't mix up the four loci

Past O/L reports show many students can't name these loci. Watch for:• "A fixed distance from a line" is two parallel lines, one on each side, not a circle and not just one line.• "Equidistant from two lines" is an angle bisector; "equidistant from two points" is a perpendicular bisector.Underline the words point or line in the question.

Construction rules

A construction uses only a straight edge (ruler) and a pair of compasses. A protractor is not allowed.• Keep the compass width fixed while you draw a matching pair of arcs.• Draw arcs long enough to cross.• Never rub out your construction arcs: they are your working, and they earn the marks.

✏️ Construct a perpendicular bisector

Draw AB = 8 cm and construct its perpendicular bisector.

1Open the compasses to more than half of AB (say 6 cm).
2With centre A, draw arcs above and below AB.
3Same width, centre B: draw arcs cutting them at X and Y.
4Join XY. It meets AB at M, the midpoint, at 90°.
5Check: AM = MB = 4 cm.

✅ XY ⊥ AB, AM = MB = 4 cm

✏️ Bisect an angle

Bisect the angle POQ.

1With centre O, draw an arc cutting the arms at P and Q.
2With centres P and Q and the same width, draw two arcs that cross at R.
3Join OR. Then ∠POR = ∠ROQ.
4Here the angle is 60°, so each half is 30°.

✅ ∠POR = ∠ROQ = 30°

Special angles without a protractor

• 60°: from O draw an arc cutting the line at P; with the same width, centre P, cut that arc at Q. ∠QOP = 60° (OPQ is equilateral).• 90°: construct the perpendicular at a point on a line.• 30° = bisect 60°. 45° = bisect 90°. 120° = two 60° angles side by side.

Two conditions? Use two loci

If a point must obey two rules, draw the locus for each rule. The point lies where the loci intersect.There may be two such points, one, or none, so draw each locus in full (the whole circle, the whole line) before you decide. Mark and label every point that fits.

⚠️ Draw the loci you're asked for

Examiners report students who mark a point by guessing, without drawing the circle of the given radius or the parallel line at the given distance. No locus drawn means no marks, even if the point is right. Draw each locus clearly, keep the arcs, and label the final point.

📌 Remember

• Fixed distance from a point → circle.• Equidistant from two points → perpendicular bisector.• Fixed distance from a line → two parallel lines.• Equidistant from two intersecting lines → angle bisector.• Two conditions → draw both loci in full; the point is where they meet. Keep all construction arcs.

🧾 Quick revision sheet

What is a locus?	The set of all points that satisfy a given condition.
Locus of points 3 cm from a point O	A circle, centre O, radius 3 cm
Locus of points equidistant from A and B	The perpendicular bisector of AB
Locus of points 2 cm from a straight line	Two lines parallel to it, 2 cm away, one on each side
Locus of points equidistant from two intersecting lines	The bisectors of the angles between them
How do you get 45° with compasses?	Construct 90° (a perpendicular), then bisect it.
Instruments allowed in a construction	Straight edge and a pair of compasses only