Geometrical constructions

The top loss is not constructing the polygon: a freehand hexagon, pentagon or octagon has unequal sides and angles, so is not regular. Equilateral triangles turn isosceles when the base is measured and apex guessed, not struck from two arcs. Fix it with compasses, each polygon in a circumscribing circle. Marks need equal sides, equal angles and visible construction.

A definition trap: thinking regular just means equal sides, so shapes get equal sides but unequal angles -- still irregular. A regular polygon needs both all sides and all angles equal: a rhombus has equal sides but is irregular, a rectangle equal angles yet irregular. Name the polygon, pick the central angle (360 over sides), and check both come out equal.

When only part of a shape is given, candidates confuse a half-hexagon with a half-octagon and construct the wrong number of sides. A half-hexagon shows three visible sides on 60 degree divisions (360 divided by 6); a half-octagon shows four on 45 degree divisions (360 divided by 8). Name the polygon and its side count first, then use the matching central angle.

When features must be spaced along a scale line (e.g. a 1 to 20 line) or a line divided proportionally, candidates measure and guess, so spacings come out uneven -- and the nets-per-sheet count too. Use the inclined-line construction: step the spaces along a faint inclined line, join the last to the far end and draw parallels to transfer them back.

Two constructions get muddled. Bisecting a line halves a length: equal arcs from each end give a perpendicular bisector at 90 degrees. Bisecting an angle halves the opening between two lines: compass on the vertex, mark a point on each arm, strike equal arcs from those and join the crossing to the vertex. Read whether a length or an angle is to be halved.

A tangent touches a circle at one point and does not enter it; a line cutting it at two points is a chord, not a tangent. At the contact point the tangent is perpendicular -- exactly 90 degrees -- to the radius there. Cutting across the circle, or meeting the radius aslant, gives a kink and loses the marks. The rule: radius to the contact point, then a right angle.

A testable error: believing a hexagon must be set out with a protractor measuring 120 degree angles. In fact a regular hexagon's side exactly equals its circumscribing radius, so setting the compass to that radius and stepping it round marks six equal points -- no protractor. Each centre segment is an equilateral triangle, so the chord equals the radius.