Class 7 Maths Chapter 14 Constructions and Tilings Visual Infographic | NCF 2023 Ganita Prakash
Chapter 6: Constructions & Tilings
Straightedge & Compass Geometry, Bisectors, Historical Rope Geometry, and Tessellations
๐ Core Constructions
Locus of all points equidistant from two endpoints. Bisects the segment at exactly 90° using intersecting compass arcs[cite: 9].
Ray dividing an angle into two equal parts[cite: 9]. Proven congruent by SSS triangle construction (ΔOBC ≅ ΔOAC)[cite: 9].
• 60°: Base of an equilateral triangle[cite: 9]
• 90°: Bisecting a straight 180° line[cite: 9]
• 30°, 45°, 15°: Formed by successive bisections[cite: 9]
๐ Replicating & Polygons
1. Copying an Angle:
Transfer base distance and cross-arc width using a compass[cite: 9]. Creates congruent triangles by SSS to ensure exact angle reproduction[cite: 9].
2. Parallel Lines via Equal Angles:
Draw a transversal through a point and copy the corresponding or alternate angle[cite: 9]. If corresponding angles are equal, lines are parallel[cite: 9].
3. Regular Hexagon (6-gon):
Composed of 6 congruent equilateral triangles meeting at a central point[cite: 9] (6 × 60° = 360°)[cite: 9]. Each interior angle is 120°[cite: 9].
๐งฉ Tiling (Tessellation)
Covering a plane completely using shapes with no gaps and no overlaps[cite: 9]. Sum of vertex angles must equal 360°[cite: 9].
Only 3 regular polygons can tile a plane alone: Equilateral Triangles (60°), Squares (90°), and Hexagons (120°)[cite: 9].
Traditional Chinese 7-piece puzzle[cite: 9] (5 triangles, 1 square, 1 parallelogram) forming endless creative geometric silhouettes[cite: 9].
๐️ Vedic Rope Geometry & Coloring Invariant Proofs
• ลulba-Sลซtras (Kฤtyฤyana c. 800 BCE): Ancient Vedic altar constructions used pegged ropes[cite: 9]. Fastening rope ends and pulling the midpoint taut above and below a line creates an exact perpendicular bisector[cite: 9].
• Architectural Arches: Trefoil and pointed arches in heritage monuments (like Red Fort, Diwan-i-Aam) are constructed using multi-centered intersecting circles and symmetry axes[cite: 9].
• The Domino Grid Rule: An m × n grid is tileable by 2 × 1 dominoes if and only if at least one dimension is even (total area is even)[cite: 9].
• Checkerboard Invariant Proof: Every 2 × 1 tile covers exactly 1 black and 1 white square[cite: 9]. If a region has unequal black and white cells, tiling is mathematically impossible[cite: 9]!
⚠️ Common Construction Traps
Trap 1: Compass Radius Too Small
When constructing a perpendicular bisector, the compass radius MUST be strictly greater than half the segment length (r > XY/2)[cite: 9]. If r is too small, the arcs will never intersect.
Trap 2: Area Even ≠ Automatically Tileable
Having an even number of total squares is a necessary condition, but NOT sufficient[cite: 9]! You must check the color balance (equal black and white squares) and boundary connectivity[cite: 9].
