Use the standard ruler and compass constructions (perpendicular bisector…

The idea

Standard constructions use a ruler and compasses only, keeping arc marks visible. To bisect a line segment, draw equal arcs from each endpoint and join the crossing points. To construct a perpendicular from or at a point on a line, draw arcs centred on the point to mark two positions, then equal arcs from those to find a second point. To bisect an angle, draw an arc across both arms from the vertex, then equal arcs from the crossing points; join the vertex to their intersection. These skills build given figures and solve loci problems. The perpendicular distance from a point to a line is the shortest distance, so a locus at a fixed distance from a line is a pair of parallel lines.

What earns the marks

• Recognises that only a ruler and compasses are permitted, with construction arcs left visible as evidence of method.
• Describes the perpendicular bisector of a line segment as the set of points equidistant from the two endpoints, found by equal arcs from each endpoint.
• Explains that constructing a perpendicular from or at a given point uses arcs centred on the point to mark two positions on the line, then equal arcs from those positions.
• States that bisecting an angle requires an arc across both arms from the vertex, then equal arcs from the two crossing points, joining the vertex to their intersection.
• Applies constructions to complete a given figure or to solve a loci problem, for example locating points equidistant from two fixed points.
• Knows that the perpendicular distance from a point to a line is the shortest distance to that line, so a locus at a fixed distance from a line is a pair of parallel lines.

Common mistakes and how to put them right

• Thinking a protractor or a measured angle may replace compass arcs.

  Put it right: The standard constructions require ruler and compasses only, so the method must be shown with arcs.

• Believing the perpendicular bisector is found by measuring the midpoint and drawing any right angle.

  Put it right: The bisector must be located by equal arcs from both endpoints.

• Assuming the shortest distance from a point to a line is measured along a slanted segment.

  Put it right: The shortest distance is the perpendicular distance, so the perpendicular must be constructed.

• Treating a locus at a fixed distance from a line as a single line or a circle.

  Put it right: It is a pair of parallel lines, one on each side of the given line.

Exam tips

• Leave every construction arc visible; rubbed-out arcs remove the evidence that the correct ruler-and-compass method was used.
• Check a completed construction by testing the defining property, for example confirming that a point on the perpendicular bisector is equidistant from both endpoints.
• For loci questions, translate the wording into a distance rule first, then decide whether the locus is a circle, a pair of parallel lines or a region bounded by arcs.