This chapter develops precise geometric constructions using only an unmarked ruler and a compass, then explores tilings of the plane. You learn to construct the perpendicular bisector of a segment (justified by congruent triangles, since any point equidistant from the two endpoints lies on it), a \(90^\circ\) angle at a point on a line, and to bisect any angle using the SSS congruence of two triangles. From these you build \(45^\circ\), \(60^\circ\), \(30^\circ\) and \(15^\circ\) angles, copy a given angle exactly, draw a line parallel to a given line, and construct a regular hexagon (made of six congruent equilateral triangles, since six \(60^\circ\) angles fill the \(360^\circ\) around a point). Ancient Shulba-Sutra rope methods are highlighted. In tiling, you ask which grids can be covered by \(2\times1\) tiles without gaps or overlaps, using a clever black-and-white colouring argument to prove some regions cannot be tiled, and you see how squares, equilateral triangles and regular hexagons can tile the entire plane.
Perpendicular bisector and 90° angleAngle bisection and copying anglesConstructing 60°, 45°, 30° angles and parallel linesConstructing a regular hexagonTiling grids and the plane
Key terms
Bisection
Dividing a line segment or angle into two identical parts.
Perpendicular bisector
A line that bisects a segment and is perpendicular to it.
Angle bisector
A ray that divides an angle into two equal angles.
Regular hexagon
A six-sided polygon with equal sides and equal angles, each \(120^\circ\).
Tiling
Covering a region using shapes without gaps or overlaps.
Compass and (unmarked) ruler
The two basic tools used for exact geometric constructions.
Extra questions & answers
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A perpendicular bisector of a segment is a line that:
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