Plane Geometry Constructions and Loci
This domain rewards method. Marks are given for the construction lines that produce the answer, not only for the finished outline, so compass arcs must be left visible and drawn thin.
Constructions you must be able to produce
- Bisect a straight line and an angle.
- Divide a line into any number of equal parts using a inclined line and parallel projection.
- Erect a perpendicular at a point on, or from a point off, a line.
- Construct regular polygons: hexagon inside a given circle, and the general method for an n-sided polygon on a given side.
- Draw tangents to a circle from an external point, and common tangents between two circles.
- Draw an ellipse by the concentric-circle or trammel method, a parabola and hyperbola from focus and directrix, and an involute, cycloid or helix.
Loci
A locus is the path of a point moving under a stated rule. Learn the standard results: equidistant from two points → perpendicular bisector; equidistant from two lines → angle bisector; fixed distance from a point → circle; fixed distance from a straight line → two parallel lines; fixed distance from a circle → two concentric circles. A point on a circle rolling along a line traces a cycloid; a point on a taut thread unwinding from a cylinder traces an involute.
Worked example
Point P stays 20 mm from a 40 mm segment AB. The locus is two straight lines parallel to AB at 20 mm, one on each side, joined by a semicircle of radius 20 mm centred on A and another centred on B. Reasoning: distance is measured to the nearest point of the segment, so beyond either end the nearest point is the endpoint itself - hence arcs, not square corners.
Traps
Rubbing out construction arcs; drawing freehand curves where plotted points are required; using the wrong radius when the locus turns a corner; leaving the answer unlabelled.