Ruler-and-compass constructions and loci – bisectors, perpendiculars and regions
MathematicsPlane GeometryAges 12–13
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Sign in to playWatch ruler-and-compass constructions animated step by step: the perpendicular bisector of a segment, the angle bisector, the perpendicular from a point to a line and at a point on a line (90°), a 60° angle, a parallel line through a point, and a regular hexagon and a square inscribed in a circle; students can also put the steps in order. The Loci screen shades points at a fixed distance from a point, a line or a segment, equidistant from two points or two lines, or meeting two conditions, with a draggable test point P.
Lesson: Ruler-and-compass constructions, shortest distance from a point to a line, loci and regions
What it shows
Every construction uses only a straight edge and a pair of compasses, and each one is justified: the perpendicular bisector and the perpendicular from a point come from a rhombus whose diagonals are perpendicular, the angle bisector from congruent triangles (SSS), the 60° angle from an equilateral triangle and the parallel line from a rhombus. The perpendicular from P meets the line at F, and PF is the shortest distance to the line. A locus is the set of points that obey a rule; regions such as closer to A than to B are bounded by loci. Distances are in centimetres on a virtual sheet.
How to use
On Constructions, choose a Construction and press Next step ▶ or ▶ Play all; ◀ Back undoes a step. Tick Put steps in order to choose each next step yourself. Drag the given points at any time and change the Compass radius. On Loci, choose a Locus, set Distance d and drag P; untick Show locus and press Mark P to find the locus yourself.
Parameters you can change
- Screen Constructions, Loci
- Construction (Constructions screen) Perpendicular bisector of a segment, Angle bisector, Perpendicular from a point to a line, Perpendicular at a point on a line (90°), 60° angle, Parallel line through a point, Regular hexagon in a circle, Square in a circle
- Compass radius (hexagon and square: radius of the circle) 1–5 cm
- Students put the steps in order
- Type of locus (Loci screen) Distance d from point A, Distance d from line l, Distance d from segment AB, Equidistant from A and B, Equidistant from two intersecting lines, Within d of A and closer to B than C
- Distance d (Loci screen) 0.5–4 cm
- Shade the region
- Show the locus
- Show the 1 cm grid
Questions to explore
- Why must the compass radius be more than half of AB to construct the perpendicular bisector?
- Why is the perpendicular PF the shortest distance from P to line l?
- Which region contains the points closer to A than to B, and what is its boundary?
