Constructions and Tilings - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Ganita Prakash II.
This compact guide covers 20 must-know concepts from Constructions and Tilings aligned with Class 7 preparation for Mathematics. Ideal for last-minute revision or daily review.
Key Points
Definition of geometric construction.
Geometric construction creates shapes using only a ruler and compass. Aimed at perfection.
Perpendicular bisector construction.
Constructed by drawing arcs from both endpoints. Points where arcs meet give the bisector.
Congruent triangles prove perpendicular bisector.
Show triangles formed during construction are congruent to confirm the bisector is valid.
To construct a 90° angle using a bisector.
Draw a perpendicular bisector at any point on a line to create a 90° angle from that point.
History in Śulba-Sūtras.
Ancient texts providing geometric construction methods for fire altars. They include basic geometric principles.
Angle bisector method.
Construct an arc from the angle's vertex; use intersection points to create equal parts, bisecting the angle.
How to construct a regular hexagon.
Using equilateral triangles, connect opposite points in a circle. Regular hexagons have all equal angles and sides.
Tiling defined.
Covering a shape's area completely using multiple copies of shapes without overlaps or gaps.
2 × 1 tiles on grids.
Check grid dimensions for tileability; only even x even grids are tileable without leftover space.
Properties of tiling.
A region can be tileable or non-tileable depending on the arrangement and count of unit squares.
Regular polygons can tile the plane.
Shapes like squares, triangles, and hexagons can cover the entire plane without gaps.
Importance of angles in tiling.
Angles must fit appropriately for effective tiling; consider sums of angles at vertices.
Point symmetry in designs.
Designs such as 6-pointed stars exhibit symmetry, using regular hexagons as a base.
Tiling in nature.
Patterns seen in beehives with hexagonal cells are examples of natural tiling effectively using space.
Copying angles.
Use a compass to transfer measurements, ensuring angles match perfectly with existing angles.
Constructing parallel lines.
Achieved by copying the angles formed by a transversal intersecting the given line.
Constructing unique arch shapes.
Various arches like trefoil depend on equal support lines and congruent angles for aesthetic appeal.
Construction of other polygons.
Using a compass and ruler, regular pentagons can be approximated, while hexagons are easily formed.
Use of ropes in ancient constructions.
Ropes served for drawing arcs, exemplifying early engineering in geometry via physical materials.
Exploring non-regular tilings.
Creative shapes can also adequately tile the plane, inspired by artists like Escher.
