OCRGCSE MathsGeometry and measuresFoundation and Higher
Visual model
A locus is a set of all possible points
ABequal distance lineperpendicular bisector
Gold-standard guide
20 mins
What you will learn
Draw accurate constructions and regions.
Use a clear step-by-step method for constructions and loci.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier geometry and measures skillsShowing clear working
Core knowledge
Know the rule, then use it
These are the short notes. Read each one, then check you can use it in the worked example below.
Method
Method
The perpendicular bisector of AB is the locus of all points equidistant from A and B
Step 1
Draw AB = 8 cm using a ruler
Mark A and B clearly
Step 2
Open compasses to more than half of AB (e.g. 5 cm) and draw arcs from A above and below the line
Both arcs must be the same radius
Step 3
Without adjusting compasses, draw arcs from B to intersect the first arcs
Label the two intersection points X and Y
Watch out
Watch out
Students rub out their construction arcs, removing evidence of method
f
Perpendicular bisector
Points on it are equally far from two fixed points.
f
Angle bisector
Points on it are equally far from two lines.
Worked example
Construct the perpendicular bisector of a line segment AB = 8 cm, then describe the locus of points equidistant from A and B.
1
Draw AB = 8 cm using a ruler: Mark A and B clearly.
2
g. 5 cm) and draw arcs from A above and below the line: Both arcs must be the same radius.
3
Without adjusting compasses, draw arcs from B to intersect the first arcs: Label the two intersection points X and Y.
4
Draw the perpendicular bisector through X and Y: This line is equidistant from A and B at every point.The locus of points equidistant from A and B is this perpendicular bisector.
Final answer
Perpendicular bisector of AB; the locus is the perpendicular bisector itself
Question ladder
Build up to the hardest questions
Do them in order. If you miss a step, read the solution, then redo the question without looking.
Workedreasoning
Construct the perpendicular bisector of a line segment AB = 8 cm, then describe the locus of points equidistant from A and B.
4 marks4 minsconstructions-and-loci-worked
Show solution
Worked solution
1.Draw AB = 8 cm using a ruler: Mark A and B clearly.
2.g. 5 cm) and draw arcs from A above and below the line: Both arcs must be the same radius.
3.Without adjusting compasses, draw arcs from B to intersect the first arcs: Label the two intersection points X and Y.
4.Draw the perpendicular bisector through X and Y: This line is equidistant from A and B at every point.The locus of points equidistant from A and B is this perpendicular bisector.
Final answer
Perpendicular bisector of AB; the locus is the perpendicular bisector itself
Mark points
M1: draw ab = 8 cm using a ruler
M1: open compasses to more than half of ab (e.g. 5 cm) and draw arcs from a above and below the line
M1: without adjusting compasses, draw arcs from b to intersect the first arcs
M1: draw the perpendicular bisector through x and y
A1: Perpendicular bisector of AB; the locus is the perpendicular bisector itself
Watch out
Students rub out their construction arcs, removing evidence of method.The examiner must see arcs to award marks — leave every arc you draw.Using a pencil and not pressing too hard makes arcs visible without being messy.
Diagnosticrecall
Describe the locus of points 3 cm from a fixed point P.
1 mark2 minsconstructions-and-loci-q1
Show solution
Worked solution
1.Spot the skill: The perpendicular bisector of AB is the locus of all points equidistant from A and B.
2.g. 5 cm) and draw arcs from a above and below the line.
3.Keep the final answer visible: A circle of radius 3 cm centred at P.
Final answer
A circle of radius 3 cm centred at P
Mark points
M1: use the correct the perpendicular bisector of ab is the locus of all points equidistant from a and b.key constructions: perpendicular bisector, angle bisector, 60° angle, perpendicular from a point to a line.always leave construction arcs visible.
A1: A circle of radius 3 cm centred at P
Watch out
Students rub out their construction arcs, removing evidence of method.The examiner must see arcs to award marks — leave every arc you draw.Using a pencil and not pressing too hard makes arcs visible without being messy.
Easyprocedure
Describe the locus of points equidistant from two parallel lines 6 cm apart.
2 marks3 minsconstructions-and-loci-q2
Show solution
Worked solution
1.Spot the skill: The perpendicular bisector of AB is the locus of all points equidistant from A and B.
2.g.5 cm) and draw arcs from a above and below the line stage first, then without adjusting compasses, draw arcs from b to intersect the first arcs.
3.Keep the final answer visible: A parallel line midway between them, 3 cm from each.
Final answer
A parallel line midway between them, 3 cm from each
Mark points
M1: use the correct the perpendicular bisector of ab is the locus of all points equidistant from a and b.key constructions: perpendicular bisector, angle bisector, 60° angle, perpendicular from a point to a line.always leave construction arcs visible.
A1: A parallel line midway between them, 3 cm from each
Watch out
Students rub out their construction arcs, removing evidence of method.The examiner must see arcs to award marks — leave every arc you draw.Using a pencil and not pressing too hard makes arcs visible without being messy.
Mediumreasoning
How do you construct a 60° angle at point A?
3 marks4 minsconstructions-and-loci-q3
Show solution
Worked solution
1.Spot the skill: The perpendicular bisector of AB is the locus of all points equidistant from A and B.
2.Use the without adjusting compasses, draw arcs from b to intersect the first arcs stage first, then draw the perpendicular bisector through x and y.
3.Keep the final answer visible: Draw a circle arc from A, then from where it crosses, draw an arc of the same radius; the intersection gives 60°.
Final answer
Draw a circle arc from A, then from where it crosses, draw an arc of the same radius; the intersection gives 60°
Mark points
M1: use the correct the perpendicular bisector of ab is the locus of all points equidistant from a and b.key constructions: perpendicular bisector, angle bisector, 60° angle, perpendicular from a point to a line.always leave construction arcs visible.
A1: Draw a circle arc from A, then from where it crosses, draw an arc of the same radius; the intersection gives 60°
Watch out
Students rub out their construction arcs, removing evidence of method.The examiner must see arcs to award marks — leave every arc you draw.Using a pencil and not pressing too hard makes arcs visible without being messy.
Hardproblem solving
Construct the angle bisector of angle ABC.
3 marks5 minsconstructions-and-loci-q4
Show solution
Worked solution
1.Spot the skill: The perpendicular bisector of AB is the locus of all points equidistant from A and B.
2.Use the draw the perpendicular bisector through x and y stage first, then draw ab = 8 cm using a ruler.
3.Keep the final answer visible: Draw equal arcs from B on both arms; from their endpoints draw equal arcs; join B to intersection.
Final answer
Draw equal arcs from B on both arms; from their endpoints draw equal arcs; join B to intersection
Mark points
M1: use the correct the perpendicular bisector of ab is the locus of all points equidistant from a and b.key constructions: perpendicular bisector, angle bisector, 60° angle, perpendicular from a point to a line.always leave construction arcs visible.
A1: Draw equal arcs from B on both arms; from their endpoints draw equal arcs; join B to intersection
Watch out
Students rub out their construction arcs, removing evidence of method.The examiner must see arcs to award marks — leave every arc you draw.Using a pencil and not pressing too hard makes arcs visible without being messy.
Exam-stylemulti-step
What is the locus of points that are less than 4 cm from A AND closer to B than to C?
4 marks6 minsconstructions-and-loci-q5
Show solution
Worked solution
1.Spot the skill: The perpendicular bisector of AB is the locus of all points equidistant from A and B.
2.g. 5 cm) and draw arcs from a above and below the line.
3.Keep the final answer visible: The region inside the circle radius 4 cm from A, on B's side of the perpendicular bisector of BC.
Final answer
The region inside the circle radius 4 cm from A, on B's side of the perpendicular bisector of BC
Mark points
M1: use the correct the perpendicular bisector of ab is the locus of all points equidistant from a and b.key constructions: perpendicular bisector, angle bisector, 60° angle, perpendicular from a point to a line.always leave construction arcs visible.
A1: The region inside the circle radius 4 cm from A, on B's side of the perpendicular bisector of BC
Watch out
Students rub out their construction arcs, removing evidence of method.The examiner must see arcs to award marks — leave every arc you draw.Using a pencil and not pressing too hard makes arcs visible without being messy.
Grade 9 stretchproblem solving
Describe the region of points that are closer to A than B and less than 3 cm from C.
A sidewithin 3 cmintersection is the answer region
4 marks7 minsloci-g9
Show solution
Worked solution
1.Use the perpendicular bisector of AB.
2.Choose the side containing A.
3.Intersect it with the inside of the circle centred at C with radius 3 cm.
Final answer
The part of the circle centred at C that lies on A's side of the perpendicular bisector of AB.
Mark points
C1: perpendicular bisector
C1: correct side
C1: intersect with circle
Watch out
Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.
Timed checkpoint
12 mins - 9 marks
Switch between skills
Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.
1Constructions and loci - 2 marksDescribe the locus of points 3 cm from a fixed point P.Mark answer
Answer
A circle of radius 3 cm centred at P
2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer
Answer
15
3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer
Answer
Rectangle, square
4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer
Answer
42 cm
Mastery check
I can explain the method for constructions and loci.
I can show clear working without skipping key steps.
I can avoid this mistake: Students rub out their construction arcs, removing evidence of method.The examiner must see arcs to award marks — leave every arc you draw.Using a pencil and not pressing too hard makes arcs visible without being messy.
This guide follows the OCR GCSE Mathematics J560 specification. Practice questions are original Learnova questions shaped around official content and exam skills.