Chapter 14: Constructions and Tilings
With just a compass and a straightedge we can halve segments and angles, copy angles, draw parallels and build hexagons, stars and arches, and we can prove each construction works using congruent triangles. Then we explore tilings: tangrams, covering grids with 2 × 1 tiles, a clever colouring proof, and which shapes can cover a whole floor.
In this chapter you will
- construct perpendicular and angle bisectors
- copy angles and draw parallel lines with a compass
- construct hexagons, stars and arches
- decide when a region or the plane can be tiled
Topics
- Geometric Constructions NCERT P2 6.1
Perpendicular bisector: equal arcs from both ends; its points are equidistant from the ends.
- Tiling NCERT P2 6.2
Tiling: cover a region with no gaps and no overlaps; 2 × 1 tiles need an even number of squares.
Key words
- Perpendicular bisector
- the line through the midpoint of a segment at 90°
- Angle bisector
- the line that splits an angle into two equal parts
- Regular hexagon
- six equal sides and six equal angles (120°)
- Tangram
- a square cut into 7 pieces to rearrange
- Tiling
- covering a region with no gaps and no overlaps
- Tessellation
- a tiling of the whole plane
Summary
- Equal arcs from two points meet on the perpendicular bisector; every point on it is equidistant from both.
- Angle bisectors, angle copies and parallels are built from equal arcs and justified by congruence.
- The radius steps 6 times round a circle: a regular hexagon with side = radius.
- 2 × 1 tiles need an even number of squares (and equal colours); a plane tiling needs 360° at every corner.