Constructions and Tilings

Chapter Overview

Geometric construction uses a ruler and compass to create exact relationships. Tiling investigates how shapes cover a surface repeatedly without gaps or overlaps.

This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.

Learning Objectives

After studying this chapter, you should be able to:

• explain and apply perpendicular bisector;
• explain and apply angle bisector and copied angle;
• explain and apply standard angles;
• explain and apply tessellations;
• communicate the reasoning behind a solution clearly;
• check answers using estimation, substitution, or a second method.

Key Concepts

| Concept | Meaning | |---|---| | Perpendicular bisector | It crosses a segment at its midpoint at 90°. Every point on it is equally distant from the segment’s endpoints. | | Angle bisector and copied angle | Compass arcs transfer equal distances, allowing an angle to be halved or reproduced accurately. | | Standard angles | A 60° angle comes from an equilateral triangle; bisecting and combining known angles produces others such as 30°, 90° and 120°. | | Tessellations | A tiling covers a plane with no gaps or overlaps. Angles meeting at a vertex must total 360°. |

Detailed Explanation

Perpendicular bisector

It crosses a segment at its midpoint at 90°. Every point on it is equally distant from the segment’s endpoints.

Angle bisector and copied angle

Compass arcs transfer equal distances, allowing an angle to be halved or reproduced accurately.

Standard angles

A 60° angle comes from an equilateral triangle; bisecting and combining known angles produces others such as 30°, 90° and 120°.

Tessellations

A tiling covers a plane with no gaps or overlaps. Angles meeting at a vertex must total 360°.

Important Rules and Formulae

• Keep the compass width unchanged when transferring an arc.
• Construction marks are part of the reasoning and should not be erased.
• Regular triangles, squares and hexagons tessellate by themselves.

Worked Examples

Example 1

Problem: Why does a regular pentagon not tessellate alone?

Solution: Its interior angle is 108°, and 360° is not a whole-number multiple of 108°.

Example 2

Problem: How is a 30° angle constructed from 60°?

Solution: Construct a 60° angle and bisect it.

Common Mistakes

• Using measurement alone where an exact compass construction is required.
• Changing compass width midway through copying an angle.
• Allowing tiles to overlap.

Quick Revision

• Perpendicular bisector: It crosses a segment at its midpoint at 90°.
• Angle bisector and copied angle: Compass arcs transfer equal distances, allowing an angle to be halved or reproduced accurately.
• Standard angles: A 60° angle comes from an equilateral triangle; bisecting and combining known angles produces others such as 30°, 90° and 120°.
• Tessellations: A tiling covers a plane with no gaps or overlaps.

Practice Questions

1. What is special about points on a perpendicular bisector?
2. How many 90° square corners meet around a point?
3. Can regular hexagons tessellate alone?
4. Which basic construction divides an angle into two equal parts?

Answers and Explanations

1. They are equidistant from the segment’s endpoints.
2. Four.
3. Yes; three 120° angles total 360°.
4. An angle bisector.

Self-Check

Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.