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Lesson 9-4. Tessellations. Transparency 9-4. D’. C’. A’. B’. 5-Minute Check on Lesson 9-3. Identify the order and magnitude of rotational symmetry for each regular polygon. 1. Triangle 2. Quadrilateral 3. Hexagon 4. Dodecagon
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Lesson 9-4 Tessellations
Transparency 9-4 D’ C’ A’ B’ 5-Minute Check on Lesson 9-3 Identify the order and magnitude of rotational symmetry for each regular polygon. 1. Triangle 2. Quadrilateral 3. Hexagon 4. 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? order: 4magnitude: 90° order: 3magnitude: 120° order: 6magnitude: 60° order: 12magnitude: 30° Standardized Test Practice: (– 4, – 2) (– 4, 2) (2, – 4) (– 2, – 4) A A B C D Click the mouse button or press the Space Bar to display the answers.
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
y x Tessellations Tessellation – a pattern using polygons that covers a plane so that there are no overlapping or empty spaces “Squares” on the coordinate plane Hexagons from many board games Tiles on a bathroom floor Not a regular or semi-regular tessellation because the figuresare not regular polygons 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
m1 Example 4-1a Determine whether a regular 16-gon tessellates the plane. Explain. Let 1 represent one interior angle of a regular 16-gon. Interior Angle Theorem Substitution Simplify. Answer: Since 157.5 is not a factor of 360, a 16-gon will not tessellate the plane.
Example 4-1b Determine whether a regular 20-gon tessellates the plane. Explain. Answer: No; 162 is not a factor of 360.
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 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. All whole numbers greater than 3 will result in a negative value for s.
Answer: There are no whole number values for n and s so that Substitution Simplify. Subtract from each side. Divide each sideby 90. Example 4-2a
Answer: No; there are no whole number values for h and s such 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.
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 486-487; 11-15, 19, 20, 26-28, 37