• Construct geometric figures using a ruler and compass.
• Create perpendicular bisectors for line segments.
• Bisect angles to create congruent angles.
• Construct specific angles such as 30°, 45°, and 60°.
• Understand the principles of tiling and how to tile regions without gaps or overlaps.
• Explore the properties of tangrams and their construction.
• Investigate the conditions for a region to be tileable.

Constructions and Tilings

Geometric Constructions

• Eyes Construction:
  • Symmetrical arcs are drawn from two centers, A and B, using a supporting line XY.
  • Condition for symmetry: AX = AY = BX = BY.

Tangrams

• Definition: Tangrams are puzzles that originated in China, made from 7 pieces obtained by dividing a square.
• Construction: Pieces can be made as cardboard cutouts or obtained from the end of the book.

Angle Construction

• Constructing 30° and 15° Angles:
  • Use a ruler and compass to bisect angles.
• Constructing a 45° Angle:
  • Bisect a 90° angle to achieve a 45° angle.

Tiling Concepts

• Definition of Tiling: Covering a region using a set of shapes without gaps or overlaps.
• Tileability Condition: A region tiled with black-and-white tiles must have an equal number of black and white tiles.

Example Problems

• Tiling the Entire Plane:
  • Squares can tile the entire plane.
• Non-tileable Regions:
  • Example: Removing a unit square from a 5 x 3 grid can make it non-tileable.