What is the locus of points exactly 4 cm from point P?
All points are the same distance from P.
The locus is a circle centred at P with radius 4 cm.
Loci and Constructions
Loci describe sets of points that satisfy specific conditions, while constructions use accurate ruler-and-compass drawing. This topic links closely to angle rules, bearings and shading regions.
Overview
A construction is an accurate drawing made using a ruler and compass.
A locus is the path traced by a point that follows a rule.
Construction = accurate geometric drawing
Locus = set of points that satisfy a condition
Locus = set of points that satisfy a condition
You need to be able to construct perpendicular bisectors, angle bisectors and accurate triangles, and recognise loci such as points equidistant from two points or two lines.
What you should understand after this topic
- Understand what constructions and loci are
- Use a ruler and compass accurately
- Construct perpendicular bisectors and angle bisectors
- Interpret common locus rules
- Shade valid regions in exam questions
Key Definitions
Construction
An accurate diagram drawn using geometrical tools such as a ruler and compass.
Locus
The path of a point that follows a particular rule.
Compass
A tool used to draw arcs and circles.
Perpendicular Bisector
A line that cuts a segment into two equal parts at \(90^\circ\).
Angle Bisector
A line that splits an angle into two equal angles.
Equidistant
The same distance from two points, lines or objects.
Region
An area that satisfies one or more locus conditions.
Arc
A curved part of a circle used during compass constructions.
Key Rules
Use only ruler and compass
Do not estimate by eye when a construction is required.
Keep arcs visible
Construction marks usually need to remain on the page.
Read the locus rule carefully
The wording tells you exactly what path to draw.
Shade the correct region
Some questions ask for the full locus, others ask for a valid area.
Important:
If a question says “less than 3 cm from P”, you usually need the region inside the circle, not just the circle boundary.
Common Locus Facts
Equidistant from two points
The perpendicular bisector of the segment joining the points.
Equidistant from two lines
The angle bisector between the lines.
Fixed distance from a point
A circle centred at that point.
Fixed distance from a line
A line parallel to the original line.
Important:
If a question says “less than 3 cm from P”, you usually need the region <strong>inside</strong> the circle, not just the circle boundary.
How to Solve
Step 1: Understand constructions and loci
Constructions are accurate diagrams made using a ruler and compass. A locus is a path or region of points that follow a rule.
Exam tip: Diagrams must be accurate, not freehand.
Keep construction arcs visible because they show your method.
Step 2: Recognise common loci
These ideas often combine with angle rules and accurate measuring.
Fixed distance from a point
Circle centred at the point.
Fixed distance from a line
Parallel lines on both sides.
Equidistant from two points
Perpendicular bisector.
Equidistant from two lines
Angle bisector.
Step 3: Perpendicular bisector
A perpendicular bisector finds points equidistant from two points.
Result: A line at \(90^\circ\) through the midpoint.
- Join the two points with a straight line.
- Place the compass at one end and draw arcs above and below the line.
- Repeat from the other end with the same compass width.
- Join the arc intersections with a straight line.
Step 4: Angle bisector
An angle bisector finds points equidistant from two lines.
Result: The angle is split into two equal parts.
- Draw an arc from the angle vertex cutting both sides.
- From each point, draw arcs inside the angle.
- Join the vertex to where the arcs meet.
Step 5: Constructing triangles
Use ruler and compass arcs to locate vertices accurately.
- Draw one side using a ruler.
- Use a compass to measure the second side.
- Draw arcs from known points.
- Join the intersection to form the triangle.
Step 6: Regions from combined conditions
Many exam questions involve more than one condition.
Exam thinking: Always draw all conditions before shading.
This is similar to the logic used in inequalities on graphs.
- Draw the first locus.
- Draw the second boundary or locus.
- Identify the overlap.
- Shade only the region that satisfies both conditions.
Step 7: Boundary vs region
Exactly
Draw only the boundary line or curve.
Less than
Shade inside the boundary.
Greater than
Shade outside the boundary.
Equidistant
Use a bisector.
Example Questions
Edexcel
Exam-style questions focusing on simple loci around points and between two points.
Edexcel
The locus is 5 cm from point P.
Describe the locus.
Edexcel
Points A and B are shown.
What is the locus of points equidistant from A and B?
AQA
Exam-style questions focusing on angle bisectors and regions within a distance.
AQA
Two lines intersect.
What is the locus of points equidistant from the two intersecting lines?
AQA
A point must be less than 4 cm from P.
What region should be shown?
OCR
Exam-style questions focusing on construction reasoning and combined locus regions.
OCR
The perpendicular bisector of AB is constructed using equal-radius arcs.
Explain why the perpendicular bisector shows points equidistant from A and B.
OCR
A point must be within 3 cm of P and closer to P than to Q.
What two boundaries are needed?
OCR
A point must be nearer to line A than line B.
Describe the boundary and the correct region.
Exam Checklist
Step 1
Read the exact wording of the construction or locus rule.
Step 2
Choose the correct construction method or locus shape.
Step 3
Leave clear compass arcs and accurate lines.
Step 4
If needed, shade only the region that satisfies all conditions.
Most common exam mistakes
Wrong bisector
Using an angle bisector instead of a perpendicular bisector, or the other way round.
Missing arcs
Rubbing out the construction evidence.
Boundary only
Drawing the line or circle but forgetting the region inside or outside.
Wrong shading
Shading a part that does not satisfy every condition.
Common Mistakes
These are common mistakes students make when working with loci and constructions in GCSE Maths.
Erasing construction arcs
Incorrect
A student removes the compass arcs after drawing the final line.
Correct
Construction arcs must be left visible as part of the method. Marks are awarded for showing how the construction was done.
Drawing by eye instead of using tools
Incorrect
A student sketches lines without using a ruler or compass.
Correct
Constructions must be accurate. Always use a compass and ruler rather than drawing freehand.
Confusing different constructions
Incorrect
A student draws a perpendicular bisector when an angle bisector is required.
Correct
Read the question carefully and identify the correct construction. A perpendicular bisector splits a line into two equal parts at 90°, while an angle bisector splits an angle into two equal angles.
Drawing only the boundary
Incorrect
A student draws the line or curve but does not show the required region.
Correct
If the question asks for a region, you must indicate it clearly, often by shading the correct side of the boundary.
Shading the wrong region
Incorrect
A student shades outside the valid area or misses the overlap.
Correct
When multiple conditions are given, only shade the region that satisfies all of them. This is usually the overlapping area.
Try It Yourself
Practise constructing loci and geometric constructions accurately.
Foundation
Foundation Practice
Recognise simple loci and standard construction rules.
What is the locus of points equidistant from two points A and B?
It cuts AB at 90° and through the midpoint.
The locus is the perpendicular bisector of AB.
Which construction is shown?
The red line cuts the segment in half at 90°.
This shows the perpendicular bisector construction.
The locus of points equidistant from two intersecting lines is called the angle ______.
It splits the angle into two equal parts.
The locus is the angle bisector.
Which line is the angle bisector?
It divides the angle into two equal parts.
The dashed blue line is the angle bisector.
A path must stay 3 m from a straight wall. What shape is the locus?
A fixed distance from a straight line makes another straight line.
The locus is a line parallel to the wall, 3 m away.
Which construction is needed to draw a 60° angle accurately?
All angles in an equilateral triangle are 60°.
Constructing an equilateral triangle gives a 60° angle.
A point must be less than 5 cm from point A. The region is inside a circle of radius how many cm?
The boundary is 5 cm from A.
The region is inside a circle of radius 5 cm.
What does a construction arc usually show?
Think compass and ruler constructions.
Construction arcs show compass marks used to build the construction accurately.
What equipment is normally used for accurate constructions?
One draws arcs; the other draws straight lines.
Use a compass and ruler for accurate constructions.
Higher
Higher Practice
Use combined loci and construction reasoning to identify exact regions.
Which shaded region shows points less than 4 cm from A and less than 4 cm from B?
The point must satisfy both conditions.
It must be inside both circles, so the answer is the overlap.
A point is equidistant from A and B, and also exactly 5 cm from A. What two loci must be used?
Equidistant from A and B gives one locus; fixed distance from A gives another.
Use the perpendicular bisector of AB and a circle centred at A with radius 5 cm.
The shaded region is inside the circle and above the line. Which description matches it?
Inside a circle means less than a fixed distance.
The region is less than a fixed distance from A and above the given line.
A park must be closer to gate A than gate B. Which side of the perpendicular bisector should be shaded?
Closer to A means all points nearer A than B.
Shade the side of the perpendicular bisector that contains A.
A point must be nearer to line l than line m. What construction helps divide the region?
Points equidistant from two intersecting lines lie on angle bisectors.
The angle bisector separates regions nearer to each line.
A region is described as more than 3 cm from P. Should the region be inside or outside the circle of radius 3 cm?
More than means farther away than the boundary.
The region is outside the circle.
Which condition describes the blue dashed line?
It is the perpendicular bisector.
The perpendicular bisector is the locus of points equidistant from A and B.
A safe zone must be at least 2 m from a wall. The boundary line is parallel to the wall. How far from the wall is the boundary?
The boundary is exactly the fixed distance away.
The boundary is 2 m from the wall.
A point must be exactly 6 cm from A and exactly 6 cm from B. Where could the point be?
Each condition creates a circle.
The point lies where the two circles intersect.
A point is less than 4 cm from P but more than 2 cm from P. What shape is the region?
It is between two circles with the same centre.
The region is an annulus: between radius 2 cm and radius 4 cm.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is a locus?
A set of points satisfying a condition.
What tools are needed?
Compass and ruler.
What is accuracy important?
Marks depend on precision.