Lesson 9-4 Tessellations

5-Minute Check on Lesson 9-3 Transparency 9-4 Click the mouse button or press the Space Bar to display the answers. Identify the order and magnitude of rotational symmetry for each regular polygon. 1. Triangle2. Quadrilateral 3. Hexagon4. Dodecagon 5. Draw the image of ABCD under a 180° clockwise rotation about the origin? 6. If a point at (-2,4) is rotated 90° counter clockwise around the origin, what are its new coordinates? Standardized Test Practice: ACBD (2, – 4) A order: 4 magnitude: 90° order: 12 magnitude: 30° order: 6 magnitude: 60° order: 3 magnitude: 120° (– 2, – 4)(– 4, 2)(– 4, – 2) D’ A’ C’ B’

Objectives Identify regular tessellations Create tessellations with specific attributes

Vocabulary Tessellation – a pattern that covers a plan by transforming the same figure or set of figures so that there are no overlapping or empty spaces Regular tessellation – formed by only one type of regular polygon (the interior angle of the regular polygon must be a factor of 360 for it to work) Semi-regular tessellation – uniform tessellation formed by two or more regular polygons Uniform – tessellation containing same arrangement of shapes and angles at each vertex

Tessellations Tessellation – a pattern using polygons that covers a plane so that there are no overlapping or empty spaces Regular Tessellation – formed by only one type of regular polygon. Only regular polygons whose interior angles are a factor of 360° will tessellate the plane Semi-regular Tessellation – formed by more than one regular polygon. Uniform – same figures at each vertex y x “Squares” on the coordinate plane Hexagons from many board games Tiles on a bathroom floor Not a regular or semi- regular tessellation because the figures are not regular polygons

Example 4-1a Determine whether a regular 16-gon tessellates the plane. Explain. Let  1 represent one interior angle of a regular 16-gon. Answer: Since is not a factor of 360, a 16-gon will not tessellate the plane. Substitution Simplify. Interior Angle Theorem m1m1

Example 4-1b Determine whether a regular 20-gon tessellates the plane. Explain. Answer: No; 162 is not a factor of 360.

Example 4-2a Determine whether a semi-regular tessellation can be created from regular nonagons and squares, all having sides 1 unit long. Solve algebraically. Each interior angle of a regular nonagon measures or 140°. Each angle of a square measures 90°. Find whole-number values for n and s such that All whole numbers greater than 3 will result in a negative value for s.

Example 4-2a Substitution Simplify. Subtract from each side. Divide each side by 90. Answer: There are no whole number values for n and s so that

Example 4-2b Determine whether a semi-regular tessellation can be created from regular hexagon and squares, all having sides 1 unit long. Explain. Answer: No; there are no whole number values for h and s such that

Example 4-3a STAINED GLASS Stained glass is a very popular design selection for church and cathedral windows. It is also fashionable to use stained glass for lampshades, decorative clocks, and residential windows. Determine whether the pattern is a tessellation. If so, describe it as uniform, regular, semi-regular, or not uniform. Answer: The pattern is a tessellation because at the different vertices the sum of the angles is 360°. The tessellation is not uniform because each vertex does not have the same arrangement of shapes and angles.

Example 4-3b STAINED GLASS Stained glass is a very popular design selection for church and cathedral windows. It is also fashionable to use stained glass for lampshades, decorative clocks, and residential windows. Determine whether the pattern is a tessellation. If so, describe it as uniform, regular, semi-regular, or not uniform. Answer: tessellation, not uniform

Summary & Homework Summary: –A tessellation is a repetitious pattern that covers a plane without overlaps or gaps –A uniform tessellation contains the same combination of shapes and angles at every vertex (corner point) Homework: –pg ; 11-15, 19, 20, 26-28, 37