5-9 Tessellations Warm Up Problem of the Day Lesson Presentation Pre-Algebra

5-9 Tessellations Warm Up Identify each polygon. Pre-Algebra 5-9 Tessellations Warm Up Identify each polygon. 1. polygon with 10 sides 2. polygon with 3 congruent sides 3. polygon with 4 congruent sides and no right angles decagon equilateral triangle rhombus

Problem of the Day If each of the capital letters of the alphabet is rotated 180° around its center, which of them will look the same? H, I, N, O, S, X, Z

Learn to predict and verify patterns involving tessellations.

Vocabulary tessellation regular tessellation semiregular tessellation

Fascinating designs can be made by repeating a figure or group of figures. These designs are often used in art and architecture. A repeating pattern of plane figures that completely covers a plane with no gaps or overlaps is a tessellation.

In a regular tessellation, a regular polygon is repeated to fill a plane. The angles at each vertex add to 360°, so exactly three regular tessellations exist.

In a semiregular tessellation, two or more regular polygons are repeated to fill the plane and the vertices are all identical.

Understand the Problem Additional Example 1: Problem Solving Application Find all the possible semiregular tessellations that use triangles and squares. 1 Understand the Problem List the important information: • The angles at each vertex add to 360°. • All the angles in a square measure 90°. • All the angles in an equilateral triangle measure 60°.

Additional Example 1 Continued 2 Make a Plan Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation. 6 triangles, 0 squares 6(60°) = 360° regular 3 triangles, 2 squares 3(60°) + 2(90°) = 360° 0 triangles, 4 squares 4(90°) = 360° regular

Additional Example 1 Continued Solve 3 There are two arrangements of three triangles and two squares around a vertex.

Additional Example 1 Continued Solve 3 Repeat each arrangement around every vertex, if possible, to create a tessellation.

Additional Example 1 Continued Solve 3 There are exactly two semiregular tessellations that use triangles and squares.

Additional Example 1 Continued Look Back 4 Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

Additional Example 2: Creating a Tessellation Create a tessellation with quadrilateral EFGH. There must be a copy of each angle of quadrilateral EFGH at every vertex.

Try This: Example 2 Create a tessellation with quadrilateral IJKL. J K L I There must be a copy of each angle of quadrilateral IJKL at every vertex.

Additional Example 3: Creating a Tessellation by Transforming a polygon Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

Additional Example 3 Continued Step 5: Use the figure to make a tessellation.

Try This: Example 3 Use rotations to create a tessellation with the quadrilateral given below. Step 1: Find the midpoint of a side. Step 2: Make a new edge for half of the side. Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side. Step 4: Repeat with the other sides.

Try This: Example 3 Continued Step 5: Use the figure to make a tessellation.

Lesson Quiz: Part 1 1. Find all possible semiregular tessellations that use squares and regular hexagons. 2. Explain why a regular tessellation with regular octagons is impossible. none Each angle measure in a regular octagon is 135° and 135° is not a factor of 360°

Lesson Quiz: Part 2 3. Can a semiregular tessellation be formed using a regular 12-sided polygon and a regular hexagon? Explain. No; a regular 12-sided polygon has angles that measure 150° and a regular hexagon has angles that measure 120°. No combinations of 120° and 150° add to 360°