pre-algebra 5.9 tessellations. identify each polygon. 1. polygon with 10 sides 2. polygon with 3...
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Identify each polygon.
1. polygon with 10 sides
2. polygon with 3 congruent sides
3. polygon with 4 congruent sidesand no right angles
decagon
equilateral triangle
rhombus
Warm Up
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.
11 Understand the Problem
Find all the possible semiregular tessellations that use triangles and squares.
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°.
Example: Problem Solving Application
22 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
Example Continued
Solve33
There are two arrangements of three triangles and two squares around a vertex.
Example Continued
Repeat each arrangement around every vertex, if possible, to create a tessellation.
Solve33
Example Continued
There are exactly two semiregular tessellations that use triangles and squares.
Solve33
Example Continued
Look Back44
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.
Example Continued
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.
Example: Creating a Tessellation
Identify each polygon.
1. polygon with 10 sides
2. polygon with 3 congruent sides
3. polygon with 4 congruent sidesand no right angles
decagon
equilateral triangle
rhombus
Warm Up
11 Understand the Problem
Find all the possible semiregular tessellations that use triangles and squares.
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°.
Example: Problem Solving Application
22 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
Example Continued
Look Back44
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.
Example Continued
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.
Example: Creating a Tessellation
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.
Example: Creating a Tessellation by Transforming a
polygon
Create a tessellation with quadrilateral IJKL.
There must be a copy of each angle of quadrilateral IJKL at every vertex.
I
J
K
L
Try This
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.
Example: 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.
Try This
Look Back44
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.
Example Continued
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.
Example: Creating a Tessellation
Identify each polygon.
1. polygon with 10 sides
2. polygon with 3 congruent sides
3. polygon with 4 congruent sidesand no right angles
decagon
equilateral triangle
rhombus
Warm Up
Example: Problem Solving Application
11 Understand the Problem
Find all the possible semiregular tessellations that use triangles and squares.
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°.
22 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
Example Continued
Look Back44
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.
Example Continued
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.
Example: Creating a Tessellation
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.
Example: Creating a Tessellation by Transforming a polygon
Create a tessellation with quadrilateral IJKL.
There must be a copy of each angle of quadrilateral IJKL at every vertex.
I
J
K
L
Try This
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.
Example: 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.
Try This
1. Find all possible semiregular tessellations that use squares and regular hexagons.
2. Explain why a regular tessellation with regular octagons is impossible. Each angle measure in a regular octagon is 135° and 135° is not a factor of 360°
none
Lesson Quiz: Part 1
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