Ruler-and-compass constructions and loci methods
Geometry and measures • Properties and constructions
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Key concepts
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Perpendicular bisector of a line segment
Definition: The perpendicular bisector is the line that is both perpendicular to a given segment and passes through its midpoint. Limitation: The construction uses only a compass and straightedge; accurate arc intersections determine the midpoint. Construction steps: From each endpoint draw arcs of equal radius greater than half the segment length. Mark the two intersection points of those arcs. Draw the line through the intersection points; that line is the perpendicular bisector and meets the segment at its midpoint.
Constructing a perpendicular to a line from a point not on the line
Definition: A perpendicular from a point to a line is a line through the point that meets the given line at a right angle. Limitation: The point may lie off the line; arcs must intersect the line twice to locate foot of perpendicular. Construction steps: With centre at the external point, draw an arc that cuts the given line at two points. With each intersection as centre and equal radius, draw arcs that intersect on the opposite side of the line. Draw the line joining the external point to the intersection of these arcs; that line meets the given line at a right angle.
Constructing a perpendicular at a given point on a line
Definition: A perpendicular at a point on a line is a line through that point making a 90° angle with the original line. Limitation: The given point must lie on the line; arc radii must be chosen so arcs intersect clearly above and below the line. Construction steps: With the given point as centre, draw arcs that cut the original line at two points equidistant from the centre. From those two intersection points draw equal-radius arcs on the same side of the line so they intersect. Draw the line through the given point and the intersection of the arcs; that line is perpendicular to the original line.
Bisecting a given angle
Definition: Angle bisection divides an angle into two equal angles. Limitation: The compass radius must be the same for arcs from the angle arms; small errors in arc placement change accuracy. Construction steps: From the angle vertex draw an arc that cuts both arms of the angle. From those two intersection points draw arcs with equal radius that intersect each other. Draw the line from the vertex to the intersection of the arcs; that line bisects the angle.
Combining constructions to build figures
Definition: Complex figures arise by combining basic constructions such as bisectors and perpendiculars. Limitation: Each step relies on previous intersection points; cumulative error increases if arcs and lines are imprecise. Application: Use perpendicular bisectors to find circumcentres, use angle bisectors to find incentres, and use perpendiculars to create right-angled constraints. Each construction produces points that become centres or vertices for further compass-and-straightedge steps.
Loci defined by ruler-and-compass constructions
Definition: A locus is the set of points satisfying a geometric condition, such as equal distance from two points or a fixed distance from a line. Limitation: Loci diagrams often combine multiple conditions; intersections of loci identify solutions. Common loci and constructions: The perpendicular bisector is the locus of points equidistant from two fixed points. A circle with centre A and radius r is the locus of points at distance r from A. The set of points at a fixed distance from a line forms two lines parallel to the given line. Combining loci determines points that satisfy multiple constraints.
Key notes
Important points to keep in mind