OCR GCSE Mathematics: Ruler-and-compass constructions and loci (J560) – Revision Notes

These revision notes cover section 8.02 (8.02a to 8.02d) of the OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026). Everything here is Foundation and Higher content; no part of it is restricted to the Higher tier. Read the study guide for the reasoning and longer examples. Test yourself with the practice questions. Tick off skills on the checklist, look at the OCR GCSE Mathematics hub, or find weak spots with the 10-minute diagnostics.

Key terms

Term	Meaning
Line segment	A straight line between two end points
Midpoint	The point exactly halfway along a segment
Bisector	A line that cuts something into two equal parts
Perpendicular	At 90° to another line
Equidistant	The same distance away from two or more things
Locus (plural loci)	All the points that follow a given rule
Perpendicular distance	The distance from a point to a line measured at 90° to the line

The four constructions at a glance

Ref	Construction	Compass work	Result
8.02a	Perpendicular bisector of AB	Same radius (more than half AB) from A and from B, arcs above and below	A line at 90° to AB through its midpoint
8.02b	Bisector of an angle	One arc from the vertex cutting both lines; then equal arcs from those two points	A line through the vertex making two equal angles
8.02c	Perpendicular from a point P to a line	Arc from P cuts the line twice; equal arcs from those two points	A line through P at 90° to the line
8.02c	Perpendicular at a point T on a line	Equal arcs cut the line either side of T; wider equal arcs from those points	A line through T at 90° to the line

Method boxes

Perpendicular bisector (also gives the midpoint)

1. Compasses wider than half of AB. Keep the setting.
2. Arcs from A (above and below AB).
3. Same radius, arcs from B to cross them.
4. Ruler through the two crossing points.
5. Midpoint M is where this line meets AB.

Angle bisector

1. Arc from the vertex V, cutting both lines at P and Q.
2. Arc from P inside the angle.
3. Same radius, arc from Q to cross it at R.
4. Ruler from V through R.

Perpendicular from a point P to a line

1. Arc from P cutting the line at A and B.
2. Arcs from A and B on the other side of the line, same radius.
3. They cross at X.
4. Join P to X.

Perpendicular at a point T on a line

1. Arcs from T cutting the line at A and B.
2. Wider arcs from A and B crossing at X.
3. Join T to X.

Must-know facts

• Every point on a perpendicular bisector is equidistant from the two end points.
• Every point on an angle bisector is equidistant from the two lines.
• The perpendicular distance from a point to a line is the shortest distance from that point to the line. Any slanted route is a hypotenuse of a right-angled triangle and is longer.
• A point is equidistant from two points if it lies on their perpendicular bisector.
• Keep your construction arcs on the page.

Loci summary

Rule	Locus	How to draw it
Fixed distance from a point	Circle	Compasses, centre the point
Within a distance of a point	Inside a circle	Circle, shade inside
Equidistant from two points	Perpendicular bisector of the segment joining them	8.02a
Equidistant from two intersecting lines	Angle bisector	8.02b
Fixed distance from a segment	Two parallel lines joined by two semicircles	Parallel lines by measuring, ends by compasses
Equidistant from two parallel lines	A parallel line half-way between	Measure the gap and halve it

Boundaries. Use a dashed line for “less than” and “more than”. Use a solid line for “at most” and “at least”. Label what your shading shows.

Method for a loci problem

1. Read the rule and decide what each condition produces: circle, bisector, parallel lines.
2. Convert real distances using the scale before you draw.
3. Draw the plan with a ruler.
4. Draw each boundary lightly, one at a time.
5. Find the region that satisfies all conditions together and shade only that.
6. Read off any measurement and convert it back using the scale.

Worked reminders

Reminder 1. A segment AB is 9.6 cm. The midpoint M is at 4.8 cm from each end.

Reminder 2. Bisecting a 58° angle gives two angles of 29°.

Reminder 3. If PF = 5 cm is the perpendicular distance and X is 12 cm along the line from F, then PX² = 25 + 144 = 169, so PX = 13 cm. The perpendicular, 5 cm, is shorter.

Reminder 4. On a plan with scale 1 cm : 4 m, a signal reaching 36 m is a circle of radius 9 cm.

Reminder 5. For all points within 3 cm of a segment 5 cm long, the outline has two straight sides of 5 cm and two semicircles of radius 3 cm, so the perimeter is 2 × 5 + 2 × π × 3 = 10 + 6π, which is 28.8 cm to 3 significant figures.

Wording to decode

Wording in a question	What to draw
“Equidistant from A and B”	Perpendicular bisector of AB
“Equidistant from the lines”	Bisector of the angle between them
“Nearer to A than to B”	The side of the perpendicular bisector that contains A
“Nearer to line m than to line n”	The side of the angle bisector that is next to m
“Not more than 5 m from X”	Solid circle, inside shaded
“More than 5 m from X”	Dashed circle, outside shaded
“Shortest distance from P to a line”	The perpendicular from P to the line
“Construct”	Compasses and ruler, with arcs left showing

Worked reminder: combining conditions

A rectangular yard has corners A, B, C, D, with AB = 8 m and AD = 5 m. Scale: 1 cm to 1 m. A bowl must be closer to A than to B, and not more than 6 m from A.

1. The perpendicular bisector of AB is the line 4 cm from A. “Closer to A” is the side containing A.
2. The other boundary is the arc of radius 6 cm centred on A. Inside it is allowed.
3. The arc meets the bisector 4.47 cm above AB, because √(6² − 4²) = √20 = 4.47.
4. The arc meets the top side CD 3.32 cm from D, because √(6² − 5²) = √11 = 3.32.
5. The region has corners A, the foot of the bisector on AB, the point 4.47 cm up the bisector, the point on CD where the arc ends, and D. One edge is curved.

Constructing figures

A figure is built from the same arcs. To construct a triangle with sides 5 cm, 6 cm and 7 cm:

1. Draw the 7 cm side XY with a ruler.
2. Compasses set to 5 cm, point on X: draw an arc above XY.
3. Compasses set to 6 cm, point on Y: draw an arc crossing it at Z.
4. Join XZ and YZ. Check that XZ = 5 cm and YZ = 6 cm.

Setting both radii to the side length gives an equilateral triangle, so each angle is 60°. Bisecting a 60° angle gives 30°, and a perpendicular gives 90°, so 90°, 45°, 60° and 30° can all be constructed with ruler and compasses.

Accuracy checklist before you move on

• Pencil sharp, lines thin.
• Compass setting unchanged within a pair of arcs.
• Arcs long enough to cross clearly.
• Scale applied before drawing.
• Final region shaded, labelled and described in words if asked.

Quick self-test

1. What is the name of the line that cuts a segment in half at 90°?
2. How wide should the compasses be when you construct a perpendicular bisector?
3. You bisect an angle of 112°. How large is each part?
4. What is the locus of points 4 cm from a fixed point?
5. What is the locus of points equidistant from two fixed points A and B?
6. What is the locus of points equidistant from two lines that cross at V?
7. A point P is 7 cm from the nearest point of a line. A student says a different point on the line is 6 cm from P. Explain why this cannot be right.
8. On a plan with scale 1 cm to 5 m, what radius do you draw for a circle of 40 m?
9. A path is 10 m long. Describe the outline of all points within 2 m of the path.
10. A point is 8 cm from a line, with foot F. X is on the line, 6 cm from F. Find the distance PX.

Answers

1. The perpendicular bisector.
2. More than half the length of the segment.
3. 56° each.
4. A circle of radius 4 cm, centre the fixed point.
5. The perpendicular bisector of AB.
6. The bisector of the angle at V (and its continuation).
7. The perpendicular distance, 7 cm, is the shortest distance, so no other point can be nearer than 7 cm.
8. 8 cm.
9. Two straight lines 2 m either side of the path, each 10 m long, joined by a semicircle of radius 2 m at each end.
10. PX² = 8² + 6² = 64 + 36 = 100, so PX = 10 cm.

Where marks are usually lost

• Arcs too short to cross, so the crossing point is a guess.
• Changing the compass radius between the two arcs of a pair.
• Rubbing out construction arcs, so there is no evidence of the method.
• Drawing the perpendicular bisector by eye instead of with arcs.
• Using a blunt pencil or thick lines, so the line is not accurate.
• Reading the scale the wrong way round.
• Shading a region that satisfies only one of the conditions.
• Drawing sharp corners instead of semicircles at the ends of a locus around a segment.
• Leaving the answer unlabelled when a question says “mark the point” or “shade the region”.

Official syllabus

OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026), section 8.02, published by OCR.