Geometry and measures - Properties and constructions: use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line

The idea

Geometric constructions use only a compass and a straight edge; a protractor must not be used. All construction arcs must be left visible. Key constructions include: the perpendicular bisector of a line segment (intersecting arcs from each end point with compass radius over half the length); bisecting an angle (an arc from the vertex, then intersecting arcs from where the first arc crosses the lines); a perpendicular from a point P to a line (an arc from P cutting the line twice, then intersecting arcs from those points); and a perpendicular at a point P on a line (arcs from P cutting the line either side, then intersecting arcs from those points). These are used to solve loci problems, which find points satisfying rules. The shortest distance from a point to a line is the perpendicular distance.

What earns the marks

• All construction arcs are left clearly visible as evidence of the method.
• Correctly selecting and executing the required construction (e.g., perpendicular bisector, angle bisector).
• Accurately constructing a perpendicular to a line from a given point, or at a given point.
• In loci problems, correctly identifying the locus from the description, constructing it, and shading the correct region if required.

Common mistakes and how to put them right

• Using a protractor to measure angles or a ruler to find a midpoint.

  Put it right: These are not constructions and will score no method marks. Use only a compass and a straight edge.

• Rubbing out construction arcs.

  Put it right: Arcs are essential working and must be left visible to be awarded marks.

• Confusing the methods for constructing a perpendicular 'from a point' versus 'at a point'.

  Put it right: For 'from a point P', the first arc must be centred on P. For 'at a point P' on the line, the first arcs are centred on P but cut the line P is on. Practise both methods to distinguish them.

Exam tips

• Use a sharp pencil and ensure your compass hinge is tight so the radius does not change accidentally during construction. Accuracy is key.
• Read loci questions very carefully. Underline key phrases like 'equidistant from', 'closer to', or 'within a certain distance of' to identify the required constructions.