What this chapter is about
This chapter teaches you how to use a ruler and compass to construct geometric shapes accurately. You will learn to draw angles of specific sizes, bisect angles and line segments, and construct triangles when certain measurements are given. These skills help you create exact figures without guessing or measuring repeatedly.
The chapter also introduces tilings, also called tessellations. A tiling is a pattern of shapes that cover a flat surface completely without gaps or overlaps. You see tilings on bathroom floors, in rangoli designs, and on fabric patterns. Understanding which shapes can tile a surface connects geometry to art and design.
After studying this chapter, you should be able to construct angles, perpendicular bisectors, and triangles using only a ruler and compass. You should also recognise which regular shapes can tile a plane and create simple tiling patterns of your own.
Key ideas
- A compass is used to draw arcs and circles of exact radius. A ruler (or straightedge) draws straight lines. Together, they let you construct precise figures.
- To bisect means to divide into two equal parts. You can bisect a line segment or an angle using arcs drawn with a compass.
- A perpendicular bisector of a line segment is a line that cuts it into two equal halves at a right angle (90°).
- To construct a triangle, you need at least three pieces of information, such as three sides (SSS), two sides and the included angle (SAS), or two angles and the included side (ASA).
- A tiling (tessellation) covers a flat surface using one or more shapes with no gaps and no overlaps.
- Regular polygons that can tile a plane by themselves are equilateral triangles, squares, and regular hexagons. Regular pentagons cannot tile a plane alone.
- The sum of angles meeting at each point in a tiling must be exactly 360°.
Formulas and facts to remember
- Sum of angles in a triangle = 180°.
- Sum of angles around a point = 360°.
- Interior angle of a regular polygon with n sides = (n − 2) × 180° ÷ n.
- Equilateral triangle interior angle = 60°. Six of them meet at a point: 6 × 60° = 360°.
- Square interior angle = 90°. Four of them meet at a point: 4 × 90° = 360°.
- Regular hexagon interior angle = 120°. Three of them meet at a point: 3 × 120° = 360°.
- Regular pentagon interior angle = 108°. This does not divide 360° evenly, so pentagons alone cannot tile a plane.
Worked examples
Example 1: Construct the perpendicular bisector of a line segment AB of length 6 cm.
Step 1: Draw a line segment AB = 6 cm using a ruler.
Step 2: Open your compass to more than half of AB (say, 4 cm). Place the compass point on A and draw arcs above and below the line.
Step 3: Without changing the compass width, place the point on B and draw arcs that cut the earlier arcs. Mark the two intersection points as P and Q.
Step 4: Use a ruler to join P and Q. This line is the perpendicular bisector. It meets AB at its midpoint M, so AM = MB = 3 cm, and angle PMB = 90°.
Example 2: Construct a triangle with sides 5 cm, 4 cm, and 3 cm (SSS construction).
Step 1: Draw a base line and mark one side, say BC = 5 cm.
Step 2: Set the compass to 4 cm. Place the point on B and draw an arc above BC.
Step 3: Set the compass to 3 cm. Place the point on C and draw an arc that cuts the first arc. Mark this intersection as A.
Step 4: Join A to B and A to C. Triangle ABC has sides AB = 4 cm, AC = 3 cm, and BC = 5 cm.
Example 3: Can regular octagons tile a floor by themselves?
Step 1: Find the interior angle of a regular octagon. n = 8, so angle = (8 − 2) × 180° ÷ 8 = 6 × 180° ÷ 8 = 1080° ÷ 8 = 135°.
Step 2: Check if 135° divides 360° evenly. 360 ÷ 135 = 2.67 (not a whole number).
Step 3: Since we cannot fit a whole number of 135° angles around a point, regular octagons alone cannot tile a plane without gaps.
Common mistakes
Changing the compass width while drawing arcs for a bisector → Keep the compass opening fixed throughout that step.
Drawing arcs that are too small to intersect → Open the compass to more than half the segment length.
Forgetting that angles at a meeting point must total 360° → Always check the angle sum when planning a tiling.
Trying to tile with regular pentagons and expecting no gaps → Remember that 108° does not divide 360° evenly; pentagons leave gaps.
Measuring with a ruler instead of constructing with compass arcs → Constructions require arcs; measuring defeats the purpose.
Quick revision
- Bisect means divide into two equal parts using compass arcs.
- Perpendicular bisector: cuts a segment at its midpoint at 90°.
- SSS, SAS, and ASA are the common ways to construct a triangle.
- Only equilateral triangles, squares, and regular hexagons tile a plane by themselves.
- Angles around any point in a tiling must add up to 360°.