Geometrical Constructions
Geometrical construction questions test compass-and-straightedge accuracy and the ability to set out polygons, curves and tangents from given data.
Constructions you must know
Bisect a line or angle with compass arcs of radius greater than half. Erect a perpendicular at a point using the 3-4-5 method or a semicircle. Divide a line into n equal parts by an inclined line, dividers and parallel lines. Construct regular polygons: hexagon from the circumcircle (radius equals side) or with the 30°/60° set square; pentagon by 72° chords. Draw an ellipse by concentric circles, intersecting arcs, or the four-centre approximate method. Draw a parabola from its focus and directrix or by the rectangle method; a cycloid by rolling a circle; an involute by unwinding a string; tangents by joining the external point to the circle centre and drawing a semicircle on it.
Worked example
Divide a 70 mm line AB into five equal parts. Draw a line from A at about 30°, step off five equal divisions with dividers, join the fifth point to B, then draw lines parallel to that join through the other divisions. Each intercept on AB is 70/5 = 14 mm. The reasoning is that parallel lines cut the two transversals proportionally, so equal steps on the inclined line produce equal steps on AB.
Traps
Using a compass radius that is too small for a bisector; not showing construction arcs; drawing parallels by eye; confusing across-flats and across-corners for a hexagon; using the axial height instead of slant height in a cone development.