Class 7 | Ganita Prakash Part-II | Chapter 6
🏗️ Constructions and Tilings
Perpendicular Bisector · Angle Bisector · Tessellations
🎨 6.1 — Geometric Constructions
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🔧 Tools for Geometric Constructions:
- Ruler — to draw straight lines
- Compass — to draw arcs and circles
- Protractor — to measure angles
- These three are enough to construct any geometric figure!
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⊥ Perpendicular Bisector
Perpendicular Bisector of a line segment XY:
A line that bisects XY at 90° (divides it into two equal halves, perpendicularly)
Construction Steps:
A line that bisects XY at 90° (divides it into two equal halves, perpendicularly)
Construction Steps:
Draw line segment XY
Set compass to more than half of XY. Draw arcs above and below from X
Same radius, draw arcs from Y. They intersect at points A and B
Join A and B — this is the perpendicular bisector!
🔑 Why does it work? (Congruence!)
Since AX = AY and BX = BY → △AOX ≅ △AOY (SSS)
Therefore OX = OY (O is midpoint) and ∠AOX = ∠AOY = 90°
Since AX = AY and BX = BY → △AOX ≅ △AOY (SSS)
Therefore OX = OY (O is midpoint) and ∠AOX = ∠AOY = 90°
🏠 Real application!
A point on the perpendicular bisector of XY is equidistant from X and Y!
Used to find a fair meeting point between two locations.
A point on the perpendicular bisector of XY is equidistant from X and Y!
Used to find a fair meeting point between two locations.
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∠ Angle Bisector
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Angle Bisector: Divides an angle into two equal angles
Draw angle ∠BAC with compass
With A as centre, draw arc cutting both rays at P and Q
With same radius, draw arcs from P and Q. They meet at R
AR is the angle bisector of ∠BAC ✅
Key Constructions to Know:
| Construction | What you need |
|---|---|
| 60° angle | Equilateral triangle base (compass only) |
| 90° angle | Perpendicular bisector at a point |
| 45° angle | Bisect the 90° angle |
| 30° angle | Bisect the 60° angle |
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🧩 6.2 — Tiling & Tessellations
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Tessellation = Tiling without gaps or overlaps!
- Regular polygons that tessellate alone: Equilateral triangle (60°), Square (90°), Regular hexagon (120°)
- Why these? Their angles add up to exactly 360° at each vertex!
- Regular pentagon (108°) does NOT tessellate alone
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Tessellation Rule:
At each meeting point (vertex), angles must add up to exactly 360°!
Triangle: 60° × 6 = 360° ✅
Square: 90° × 4 = 360° ✅
Hexagon: 120° × 3 = 360° ✅
Pentagon: 108° × ? = not 360° ❌
At each meeting point (vertex), angles must add up to exactly 360°!
Triangle: 60° × 6 = 360° ✅
Square: 90° × 4 = 360° ✅
Hexagon: 120° × 3 = 360° ✅
Pentagon: 108° × ? = not 360° ❌
🏛️ Tessellation in Nature & Architecture!
Honeycomb (hexagons), bathroom tiles, Islamic geometric art, Red Fort arches, football (pentagons + hexagons together!), fish scales, snake skin...
Honeycomb (hexagons), bathroom tiles, Islamic geometric art, Red Fort arches, football (pentagons + hexagons together!), fish scales, snake skin...
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📝 Quick Summary
Perp. bisector:
90° + midpoint
of line segment
90° + midpoint
of line segment
Angle bisector:
divides angle
into 2 equal parts
divides angle
into 2 equal parts
Tessellate if:
angles at vertex
add to 360°
angles at vertex
add to 360°
Triangle, Square,
Hexagon tessellate!
Pentagon doesn't!
Hexagon tessellate!
Pentagon doesn't!
🌟 Chapter 6 Complete! 🌟
Made with ❤️ by @edugrown
Made with ❤️ by @edugrown