Class 7 | Ganita Prakash Part-II | Chapter 6
๐๏ธ Constructions and Tilings
Perpendicular Bisector ยท Angle Bisector ยท Tessellations
๐จ 6.1 โ Geometric Constructions
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๐ง Tools for Geometric Constructions:
- Ruler โ to draw straight lines
- Compass โ to draw arcs and circles
- Protractor โ to measure angles
- These three are enough to construct any geometric figure!
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โฅ Perpendicular Bisector
Perpendicular Bisector of a line segment XY:
A line that bisects XY at 90ยฐ (divides it into two equal halves, perpendicularly)
Construction Steps:
A line that bisects XY at 90ยฐ (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 โ this is the perpendicular bisector!
๐ Why does it work? (Congruence!)
Since AX = AY and BX = BY โ โณAOX โ โณAOY (SSS)
Therefore OX = OY (O is midpoint) and โ AOX = โ AOY = 90ยฐ
Since AX = AY and BX = BY โ โณAOX โ โณAOY (SSS)
Therefore OX = OY (O is midpoint) and โ AOX = โ AOY = 90ยฐ
๐ 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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โ Angle Bisector
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Angle Bisector: Divides an angle into two equal angles
Draw angle โ 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 โ BAC โ
Key Constructions to Know:
| Construction | What you need |
|---|---|
| 60ยฐ angle | Equilateral triangle base (compass only) |
| 90ยฐ angle | Perpendicular bisector at a point |
| 45ยฐ angle | Bisect the 90ยฐ angle |
| 30ยฐ angle | Bisect the 60ยฐ angle |
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๐งฉ 6.2 โ Tiling & Tessellations
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Tessellation = Tiling without gaps or overlaps!
- Regular polygons that tessellate alone: Equilateral triangle (60ยฐ), Square (90ยฐ), Regular hexagon (120ยฐ)
- Why these? Their angles add up to exactly 360ยฐ at each vertex!
- Regular pentagon (108ยฐ) does NOT tessellate alone
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Tessellation Rule:
At each meeting point (vertex), angles must add up to exactly 360ยฐ!
Triangle: 60ยฐ ร 6 = 360ยฐ โ
Square: 90ยฐ ร 4 = 360ยฐ โ
Hexagon: 120ยฐ ร 3 = 360ยฐ โ
Pentagon: 108ยฐ ร ? = not 360ยฐ โ
At each meeting point (vertex), angles must add up to exactly 360ยฐ!
Triangle: 60ยฐ ร 6 = 360ยฐ โ
Square: 90ยฐ ร 4 = 360ยฐ โ
Hexagon: 120ยฐ ร 3 = 360ยฐ โ
Pentagon: 108ยฐ ร ? = not 360ยฐ โ
๐๏ธ 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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๐ Quick Summary
Perp. bisector:
90ยฐ + midpoint
of line segment
90ยฐ + 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 360ยฐ
angles at vertex
add to 360ยฐ
Triangle, Square,
Hexagon tessellate!
Pentagon doesn't!
Hexagon tessellate!
Pentagon doesn't!
๐ Chapter 6 Complete! ๐
Made with โค๏ธ by @edugrown
Made with โค๏ธ by @edugrown