Ruler-and-compass constructions

OCR J560 awards method marks for "construction lines" — the faint arcs that prove you used a compass. Don't rub them out: marks come from showing the construction, not just the final line.

Constructions you must know

1. Perpendicular bisector of a line segment AB

1. Open the compass to more than half AB.
2. Place the point at A, draw arcs above and below AB.
3. Without changing the compass, place the point at B and draw arcs above and below.
4. The two pairs of arcs cross at two points; draw a straight line through them. This line is perpendicular to AB and bisects it.

The perpendicular bisector is the locus of points equidistant from A and B.

2. Bisecting an angle

1. Place compass at the vertex of the angle; draw an arc that crosses both arms.
2. From each crossing point, with an unchanged compass radius, draw an arc inside the angle.
3. The two new arcs cross at one point; draw a line from the vertex through this point — the angle bisector.

The angle bisector is the locus of points equidistant from the two arms.

3. Perpendicular from a point P to a line L

1. Place compass at P; draw an arc crossing L at two points (M and N).
2. From M and N, with the same radius, draw arcs that cross on the opposite side of L.
3. Join P to that crossing — perpendicular to L.

4. Constructing 60° angles

Use the compass to draw an equilateral triangle. All angles are 60°. Combining bisectors gives 30° and 15°.

Loci (related skill)

A locus is the set of all points satisfying a given condition.

Common loci:

• Equidistant from a single point: a circle.
• Equidistant from two points: the perpendicular bisector of the line joining them.
• Equidistant from two intersecting lines: the angle bisectors.
• Within a fixed distance r of a point: a disk (filled circle) of radius r.
• Within a fixed distance r of a line segment: a stadium shape (two semicircles + rectangle).

OCR mark scheme conventions

• M1 for correct construction lines visible (arcs from both endpoints).
• A1 for the correct final line drawn.
• "Show your construction lines clearly" appears explicitly in the question — don't erase them.
• Lengths within ±2 mm; angles within ±2°.