Loci & Constructions | FractionRush

Welcome to Loci & Constructions!

A locus (plural: loci) is the set of all points satisfying a given condition. Constructions use only a compass and straightedge to draw exact geometric figures. Together, these skills are essential for IGCSE exams — expect to draw perpendicular bisectors, angle bisectors, and shade regions defined by multiple loci.

Locus = the path traced by a point obeying a rule | Always use a sharp compass and ruler

Perpendicular Bisector

Equal distance from two points

Angle Bisector

Equal distance from two lines

Locus — Fixed Distance

Circle around a point

Locus — From a Line

Parallel lines + semicircles

Region Problems

Shade areas satisfying multiple conditions

Scale & Bearings

Apply loci to real-world maps

1. Perpendicular Bisector of a Line Segment

The perpendicular bisector of AB is the line that is perpendicular to AB and passes through its midpoint. Every point on this line is equidistant from A and B.

Construction steps (compass & ruler only):
1. Open your compass to more than half the length of AB.
2. Place the compass on point A and draw an arc above and below AB.
3. Without changing the compass width, place it on point B and draw two more arcs — they should intersect the first pair.
4. Mark the two intersection points (call them P and Q).
5. Draw a straight line through P and Q using your ruler.
6. This line PQ is the perpendicular bisector of AB.

Always leave your construction arcs visible in an exam — they show your method and earn marks even if the final line is slightly off.

2. Angle Bisector

The angle bisector divides an angle into two equal halves. Every point on the bisector is equidistant from the two arms of the angle.

Construction steps:
1. Place the compass on the vertex (V) of the angle. Draw an arc that crosses both arms of the angle. Label the crossing points C and D.
2. Place the compass on point C and draw an arc inside the angle.
3. Without changing the compass width, place it on point D and draw another arc. These two arcs should intersect — label that point E.
4. Draw a straight line from V through E.
5. VE is the angle bisector.

Keep the compass width the same for steps 2 and 3 — this guarantees CE = DE and ensures the bisector is accurate.

3. Perpendicular from a Point to a Line

This drops a perpendicular from an external point P down to a line l. The foot of the perpendicular is the closest point on l to P.

Construction steps:
1. Place the compass on P. Open it wide enough to draw an arc that crosses the line l at two points. Label these A and B.
2. Without changing width, place the compass on A and draw an arc on the opposite side of the line from P.
3. Same width, place on B and draw another arc. Label the intersection Q.
4. Draw the line PQ — this is perpendicular to l.

4. Perpendicular at a Point on a Line

This constructs a line perpendicular to l at a point P that already lies on l.

Construction steps:
1. Place the compass on P. Mark two points A and B on the line at equal distances from P (one each side).
2. Increase the compass width. Place on A and draw an arc above the line.
3. Same new width, place on B and draw another arc above the line. Label the intersection Q.
4. Draw line PQ — perpendicular to l at P.

5. Locus — Fixed Distance from a Point

Locus of points at distance r from fixed point P = a circle, centre P, radius r

All points exactly 3 cm from P lie on a circle of radius 3 cm centred at P.
Points less than 3 cm from P → inside the circle.
Points more than 3 cm from P → outside the circle.

Draw the full circle with a compass. In region problems, the circle boundary is included (solid line) for ≤ or ≥, and excluded (dashed line) for < or >.

6. Locus — Equidistant from Two Points

Locus of points equidistant from A and B = perpendicular bisector of AB

If PA = PB, then P lies on the perpendicular bisector of segment AB.
Construct this using the perpendicular bisector method from Section 1.

This is the single most tested locus in IGCSE. Practise constructing it until it takes under 60 seconds.

7. Locus — Equidistant from Two Lines

Locus of points equidistant from two intersecting lines = the angle bisectors of the angles between the lines

There are two angle bisectors (they are perpendicular to each other). Together they give all points equidistant from both lines.
For a single angle (e.g. the acute angle), draw just the bisector of that angle.
Construct using the angle bisector method from Section 2.

For parallel lines, the locus equidistant from both is the parallel line exactly halfway between them.

8. Locus — Fixed Distance from a Line Segment

Locus of points at distance d from line segment AB = two parallel lines (distance d from AB) joined by semicircles of radius d at each endpoint

The locus has three parts:
1. A line parallel to AB, distance d above it.
2. A line parallel to AB, distance d below it.
3. A semicircle of radius d centred at A (closing the left end).
4. A semicircle of radius d centred at B (closing the right end).
Together these form a "stadium" or "discorectangle" shape.

For an infinite line (not a segment), the locus is just two parallel lines — no semicircles needed at the ends.

9. Region Problems — Combining Loci

Exam questions often ask you to shade the region satisfying two or more conditions simultaneously. Work through each condition separately, then identify the overlap.

Typical phrasing:
"Shade the region that is closer to A than to B, AND within 4 cm of C."
Method:
1. Draw the perpendicular bisector of AB (boundary for "closer to A").
2. Draw a circle of radius 4 cm centred at C.
3. The required region is on the A-side of the bisector AND inside the circle.

Inequalities and loci:
Closer to A than B → on the A-side of the perpendicular bisector of AB.
Closer to line PQ than line RS → on the PQ-side of the angle bisector.
Within d of point P → inside circle of radius d at P.
More than d from line l → outside the parallel lines at distance d.

Always construct all boundaries first. Only shade after all lines/circles are drawn. Use light shading (hatching) so your constructions remain visible.

10. Scale Drawings and Bearings with Loci

Real-world problems give distances in km or miles, and a scale (e.g. 1 cm = 2 km). Convert all distances to cm before drawing.

Key formulae:
Drawing length = Real length ÷ Scale denominator
Real length = Drawing length × Scale denominator

Bearings: Always measured clockwise from North (000°–360°).
North is straight up. East = 090°, South = 180°, West = 270°.

Example approach: Town A is 6 km from town B on a bearing of 070°. At scale 1 cm : 2 km, draw B at distance 3 cm from A, at 70° clockwise from North.

In bearing problems, always draw a North arrow at the starting point before measuring the bearing angle.

Example 1 — Perpendicular Bisector Construction

Given: Points A and B are 6 cm apart. Construct the perpendicular bisector of AB.

Step 1: Open compass to 4 cm (more than half of 6 cm). Place on A, draw arcs above and below the line AB.

Step 2: Keep compass at 4 cm. Place on B, draw two more arcs crossing the first pair at points P (above) and Q (below).

Step 3: Draw line PQ. This crosses AB at its midpoint (3 cm from each end) at exactly 90°.

Locus meaning: Every point on line PQ is exactly equidistant from A and B.

Example 2 — Angle Bisector of 60°

Given: Angle ABC = 60°. Bisect this angle.

Step 1: Compass on B at any convenient radius. Draw arc cutting BA at C and BC at D.

Step 2: Compass on C (same radius), draw arc inside angle. Compass on D (same radius), draw arc — intersects first at E.

Step 3: Draw ray BE. Angle ABE = angle EBC = 30°. BE is the bisector.

Key property: Any point on BE is equidistant from lines BA and BC.

Example 3 — Locus from a Line Segment

Given: Line segment PQ = 5 cm. Draw the locus of all points exactly 2 cm from PQ.

Step 1: Draw a line parallel to PQ, 2 cm above it (length 5 cm, matching PQ).

Step 2: Draw a line parallel to PQ, 2 cm below it.

Step 3: At end P, draw a semicircle of radius 2 cm connecting the two parallel lines.

Step 4: At end Q, draw another semicircle of radius 2 cm connecting the two parallel lines.

Result: A stadium shape. Total width (across the segment) = 5 cm; total height = 4 cm; curved caps have radius 2 cm.

Example 4 — Region Problem (Two Conditions)

Given: Points A and B are 8 cm apart. Point C is 5 cm from A and 6 cm from B. Shade the region: within 3 cm of C AND closer to A than to B.

Step 1: Construct the perpendicular bisector of AB. This is the boundary for "equidistant from A and B".

Step 2: Draw a circle of radius 3 cm centred at C.

Step 3: Identify the A-side of the perpendicular bisector (the half containing A).

Step 4: Shade the area that is BOTH inside the 3 cm circle AND on the A-side of the bisector.

Exam tip: Describe the region in words: "inside the circle centre C radius 3 cm, on the same side of the perpendicular bisector of AB as A".

Example 5 — Scale Drawing with Bearings

Given: A ship leaves port P and sails 30 km on a bearing of 120° to reach port Q. Scale: 1 cm = 5 km. Draw PQ and find the locus of all points within 15 km of P.

Step 1: Convert 30 km → 30 ÷ 5 = 6 cm. Convert 15 km → 15 ÷ 5 = 3 cm.

Step 2: Draw a North arrow at P. Measure 120° clockwise from North. Draw PQ = 6 cm in that direction.

Step 3: Draw a circle of radius 3 cm centred at P. This represents all points within 15 km of P.

Check: Q is at 6 cm from P. Since 6 cm > 3 cm, port Q lies outside the 15 km zone — it is more than 15 km from P. ✓

Example 6 — Three-Condition Region

Given: In triangle ABC: AB = 7 cm, BC = 6 cm, AC = 5 cm. Shade the region inside the triangle that is: (i) closer to AB than to BC, (ii) within 3 cm of B, (iii) closer to A than to C.

Step 1 — Condition (i): Bisect angle ABC. Points on the B-side (towards AB) of this bisector are closer to AB.

Step 2 — Condition (ii): Draw a circle radius 3 cm centred at B.

Step 3 — Condition (iii): Construct perpendicular bisector of AC. Points on the A-side are closer to A than to C.

Step 4: Shade the region inside triangle ABC that satisfies ALL THREE conditions simultaneously — this is the intersection of all three regions.

Interactive Construction Visualiser

Select a construction type, then click on the canvas to set points. Watch the construction appear step by step.

Construction:

Click on the canvas to place points, or press "Show All" to see the full construction.

Locus Explorer — Click any point

The canvas below shows points A and B fixed. Click anywhere on the canvas to see your clicked point's distance from A, distance from B, and which locus it is closest to.

Click anywhere on the canvas above to analyse that point's locus properties.

Exercise 1 — Perpendicular Bisector & Distance from a Point

Answer with a number (distances in cm, angles in degrees). Use the locus rules you have learned.

1. A and B are 10 cm apart. A point P lies on the perpendicular bisector of AB. How far is P from A if P is also 6 cm from the midpoint of AB? (Enter distance in cm — use Pythagoras.)

2. The perpendicular bisector of AB passes through the midpoint of AB. AB = 14 cm. How far from A is the midpoint? (cm)

3. In a perpendicular bisector construction, you open your compass to more than half of AB. AB = 8 cm. What is the minimum compass opening (in cm) you should use? (Enter the boundary value.)

4. Point P is equidistant from A and B where AB = 12 cm. P lies 10 cm from A. How far is P from B?

5. A locus is described as "the set of all points exactly 5 cm from point O". What is the name of this shape? [Circle=1, Square=2, Triangle=3]

6. How many intersection points do the two arcs in a perpendicular bisector construction make? (Enter the number.)

7. The angle between the perpendicular bisector and the original segment AB is how many degrees?

8. Point Q is on the perpendicular bisector of AB. AB = 6 cm. Q is 5 cm from the midpoint M of AB. How far is Q from A? (Use Pythagoras: QA² = QM² + AM²)

9. You draw a circle of radius 4 cm centred at P to represent a locus. A point X is 6 cm from P. Is X inside (enter 0) or outside (enter 1) the locus circle?

10. AB = 10 cm. A point P is equidistant from A and B and lies 13 cm from A. How many cm is P from the line AB? (i.e. the perpendicular distance — use Pythagoras with AM = 5 cm, PA = 13 cm.)

Exercise 2 — Angle Bisector & Locus from Two Lines

Answer numerically. All angles in degrees.

1. An angle of 80° is bisected. What is each half-angle?

2. An angle of 50° is bisected. What is each half-angle?

3. The angle bisector of a 90° angle makes what angle with each arm?

4. Two angle bisectors of the angles at a point are always perpendicular to each other. If they bisect supplementary angles (summing to 180°), what is the angle between the two bisectors?

5. An angle of 120° is bisected. Each half is how many degrees?

6. A point P is equidistant from two lines that form a 60° angle. P lies on the bisector of that angle. The bisector makes what angle with each line?

7. When constructing an angle bisector, how many arcs do you draw with the compass placed at the vertex? (Enter the number.)

8. Two roads cross at 40°. A new path is built equidistant from both roads. What angle does the path make with each road?

9. Bisecting a 90° angle gives 45°. Bisecting that 45° gives how many degrees?

10. The locus equidistant from two parallel lines is a line parallel to both, midway between them. If the lines are 8 cm apart, the locus line is how many cm from each?

Exercise 3 — Locus at Fixed Distance from a Line Segment

Give numerical answers. All distances in cm, all areas in cm².

1. A segment PQ = 6 cm. The locus of points 2 cm from PQ (a "stadium") has two straight edges on the long sides. How long is each straight edge? (same as PQ length)

2. Same stadium as Q1. The semicircle caps have radius how many cm?

3. A segment AB = 10 cm. Points exactly 3 cm from AB form a stadium. The total length of the stadium (end to end, measured along the long axis) is AB + 2r. Calculate this total length. (cm)

4. A segment of length 5 cm. Locus at distance 4 cm. The width of the stadium perpendicular to the segment is 2r. What is this width? (cm)

5. The area of a stadium = (length of segment × 2r) + π r². For segment 6 cm and r = 2 cm, calculate the area to 1 d.p. (cm²) Use π ≈ 3.142.

6. For an infinite line (not a segment), the locus at distance d is just two parallel lines. How many parallel lines form this locus?

7. A segment 8 cm long. Locus 1.5 cm away. How many cm long are the two parallel straight edges of the stadium?

8. For the stadium in Q7, the total perimeter = 2 × segment length + 2πr. Calculate to 1 d.p.

9. A corridor of width 4 cm is formed by points within 2 cm of a line segment. This is the same as saying the corridor is how many cm wide? (total width = 2r)

10. Two points A and B are 12 cm apart. Locus at 5 cm from A = circle radius 5 cm. Locus at 5 cm from B = circle radius 5 cm. These two circles intersect. The perpendicular bisector of AB passes through both intersection points. How far from AB's midpoint are the intersections? (Use Pythagoras: d² = 5² − 6²) Enter d in cm.

Exercise 4 — Region Problems

Numerical answers about regions defined by multiple loci conditions.

1. A and B are 6 cm apart. How many conditions are needed to define the region "closer to A than to B AND within 4 cm of A"?

2. A point is "closer to A than to B". The boundary of this region is the perpendicular bisector of AB. How many regions does this bisector divide the plane into?

3. A triangle has vertices P, Q, R. The region "closer to PQ than to PR" is bounded by the bisector of angle P. If angle P = 70°, each side of the bisector spans how many degrees?

4. Two loci boundaries (a circle and a straight line) divide the plane into at most how many separate regions?

5. A circle (radius 3 cm) and the perpendicular bisector of AB create regions. A point is inside the circle AND on the A-side of the bisector. How many separate areas satisfy this? (Assume bisector cuts through the circle.) Enter 1 or 2.

6. In a region problem "within 4 cm of P and within 5 cm of Q" where PQ = 7 cm, the two circles overlap. The region required is the intersection. How many arcs bound this intersection region?

7. A region is described as "more than 3 cm from point P". Is the boundary circle included (enter 1) or excluded (enter 0) from the region?

8. In a triangle ABC, the set of points equidistant from all three sides is the incentre. It lies on the intersection of how many angle bisectors? (minimum number needed)

9. In a triangle ABC, the set of points equidistant from all three vertices is the circumcentre. It lies on the intersection of how many perpendicular bisectors? (minimum)

10. Points closer to AB than to AC lie on the side of the angle bisector of A that contains AB. If the full angle at A is 80°, the "closer to AB" region spans how many degrees at A?

Exercise 5 — Scale Drawings and Bearings with Loci

Convert between real distances and drawing distances. Bearings are 3-digit, measured clockwise from North.

1. Scale 1 cm : 2 km. A real distance of 10 km is how many cm on the drawing?

2. Scale 1 cm : 5 km. A drawing distance of 4 cm represents how many km?

3. Scale 1 : 50 000. A drawing distance of 3 cm represents how many metres in real life? (1 cm = 50 000 cm = 500 m)

4. A bearing of 090° points in which compass direction? [North=1, East=2, South=3, West=4]

5. A bearing of 270° points in which compass direction? [North=1, East=2, South=3, West=4]

6. A ship sails from A to B on bearing 060°. On a 1:100 000 map the distance AB is 6 cm. What is the real distance in km?

7. Scale 1 cm : 3 km. A lighthouse is 9 km from the harbour. On the diagram it is how many cm?

8. A locus of "within 6 km of town P" is drawn at scale 1 cm : 2 km. The locus circle on the diagram has radius how many cm?

9. Bearing of South-West is 225°. The back-bearing (reverse bearing) from B to A is 225° − 180° = ? degrees.

10. At scale 1 cm : 4 km, two towns are 5 cm apart on the map. What is the real distance in km?

Practice — 20 Mixed Questions

Mixed loci and constructions questions. Read carefully and enter numerical answers.

1. A and B are 8 cm apart. A point on the perpendicular bisector of AB is 5 cm from the midpoint. How far is it from A? (Pythagoras: 4² + 5²)

2. An angle of 70° is bisected. Each half = ?°

3. A circle locus has centre P and radius 7 cm. A point 9 cm from P is inside (0) or outside (1) the circle?

4. Stadium locus: segment 5 cm, distance 3 cm. Total end-to-end length = 5 + 2(3) = ?

5. Stadium locus perimeter = 2 × 5 + 2π × 3. Calculate to nearest whole number. (π ≈ 3.142)

6. Scale 1 cm : 5 km. Real distance 35 km → ? cm on diagram.

7. Bearing 180° points which direction? [N=1,E=2,S=3,W=4]

8. The perpendicular bisector of AB always passes through the _____ of AB. [midpoint=1, endpoint=2, centroid=3]

9. Two lines cross at 80°. Angle bisector makes what angle with each line?

10. A point is equidistant from A and B. If PA = 7 cm, then PB = ? cm

11. Locus of points 4 cm from a line (infinite): how many parallel lines form this locus?

12. Circumcentre of a triangle is equidistant from all three _____. [vertices=1, sides=2, midpoints=3]

13. Incentre of a triangle is equidistant from all three _____. [vertices=1, sides=2, midpoints=3]

14. Scale 1:20000. A map distance of 8 cm = ? metres in real life. (8 × 20000 cm = ? m)

15. A and B are 10 cm apart. Locus equidistant from A and B is the perpendicular bisector, which passes 5 cm from each. A point P on the bisector is 13 cm from A. Its perpendicular distance from AB = ? cm (Pythagoras: 13² − 5²)

16. Two circles (same radius 5 cm) are centred at A and B, 6 cm apart. Do they intersect? [Yes=1, No=0] (Each radius 5 > half of 6 = 3, so yes.)

17. Back-bearing: if bearing from A to B is 050°, bearing from B to A = 050 + 180 = ?°

18. How many arcs are needed in total to construct the perpendicular bisector of AB? (2 from A + 2 from B = ?)

19. Angle bisector construction: you place compass at vertex, then at the two arc-crossing points. How many times do you reposition the compass on the arms? (Enter 2.)

20. A locus "within 3 cm of P AND within 4 cm of Q" where PQ = 6 cm. The two circles overlap. The number of boundary arcs forming the required region = ?

Challenge — 8 Harder Problems

Region problems, scale drawing, and multi-step loci. Show all working on paper before entering your answer.

1. Triangle ABC has AB = 8 cm, angle A = 60°, angle B = 70°. A point P inside the triangle is equidistant from A and B, and equidistant from sides AB and BC. P is at the intersection of the perpendicular bisector of AB and the angle bisector of B. Angle B = 70°; the bisector makes what angle with BC? (Enter the angle in degrees.)

2. Two towns A and B are 24 km apart. A helicopter must stay within 15 km of A and within 18 km of B. Using scale 1 cm : 3 km, the circle for A has radius ? cm on the diagram.

3. In the same problem (Q2), the circle for B has radius ? cm on the diagram.

4. A ship leaves port P on bearing 040° and travels 50 km to port Q. Scale 1 cm : 10 km. PQ on the diagram = ? cm.

5. The back-bearing from Q to P in question 4 = 040° + 180° = ?°.

6. A garden is rectangular, 12 m × 8 m. A sprinkler at one corner waters all points within 10 m of that corner. Area of garden within range = area of quarter-circle (radius 10, but constrained to rectangle). Quarter-circle area = π × 10² / 4 ≈ ? m². (π ≈ 3.142, to nearest whole number.)

7. Points equidistant from the three vertices of a triangle are found at the circumcentre. For an equilateral triangle with side 6 cm, the circumradius = side / √3 ≈ 6 / 1.732 ≈ ? cm. (To 1 d.p.)

8. A and B are 10 cm apart. Shade the region: closer to B than to A, AND more than 4 cm from A, AND within 7 cm of B. The region is bounded by 3 curves/lines. How many separate closed boundaries form the shaded region? (Enter 3 for the 3 boundary components.)