Chapter at a Glance – Constructions and Tilings

This chapter focuses on exact geometric constructions using only ruler and compass, symmetry, special angles, regular polygons, and tiling concepts.

Main Topics Covered

• Perpendicular bisector and its properties
• Constructing 90°, 60°, 45°, 30°, 15° angles
• Angle bisector and copying an angle
• Constructing parallel lines
• Regular hexagon and 60° angle via equilateral triangle
• Arch designs and symmetric figures
• Tiling regions and the plane (regular polygons)

Key Takeaways for Exams

Key Construction Steps – Ruler & Compass

Step-by-step methods (most frequently asked in exams).

1. Perpendicular Bisector of Segment XY

• Arc from X (radius > half XY), arc from Y same radius – intersect at A (above).
• Same below – intersect at B.
• Join AB: perpendicular bisector.

2. Angle Bisector of ∠AOB

• Arc centred at O, mark equal points A', B' on arms.
• Equal arcs from A' and B' – intersect at C.
• OC is bisector.

3. 60° Angle (Equilateral Triangle)

• Arc from A on line AX.
• Intersect with arc from intersection point B (same radius) at C.
• ∠CAX = 60°.

4. Regular Hexagon (side length = radius)

• Circle centre O, radius = side.
• From point on circle, mark arcs of same radius around circle – 6 points.
• Join consecutively.

5. Parallel Line through Point B to Line m

• Transversal through B intersecting m at A.
• Copy alternate/corresponding angle at B.

Key Property

• Points equidistant from X and Y lie on perpendicular bisector of XY.
• Congruence (SSS, SAS) used for justifications.

Concept Cards – Quick Explanations

Examples + Construction Steps

Eye Construction (Symmetry)

Pointed Arch

6-Pointed Star

Parallel Lines

Figure it Out Solutions (All Explained)

Perpendicular Bisector Questions

Angle Bisection

Tiling

Extra Practice Questions (Exam-Ready) – Chapter 6

Questions • Steps, Justifications, Constructions • With Solutions

1-Mark Questions

2-Mark Questions

3-Mark Questions (Justify)

4–5 Mark Questions (Construction)

Challenge (Tiling)

Common Mistakes & How to Avoid

History & Fun Facts

Ancient India: Śulba-Sūtras

Rope-based constructions for altars (perpendiculars, bisectors).

Real-Life

• Architecture: Arches, domes.
• Tiles: Floors, beehives (hexagonal).
• Art: Escher tilings.

Fun Facts

• Only 3 regular polygons tile plane.
• Bees use hexagons (optimal space).
• Recent semi-regular tilings discovered (2023).

Quick Revision One-Pager

Must-Know Constructions

• ✓ Perp Bisector: Equal arcs from ends.
• ✓ 90°: Perp bisector through point.
• ✓ Angle Bisect: Equal points + arcs.
• ✓ 60°: Equilateral △.
• ✓ Hexagon: 6 × 60° arcs.
• ✓ Parallel: Copy angle.

Exam Tips

Interactive Quiz – 15 Questions