9-6 Compositions of Reflections

Quick Review

The diagram shows a glide reflection of N. A glide reflection is an isometry in which a figure and its image have opposite orientations.

There are exactly four isometries: translation, reflection, rotation, and glide reflection. Every isometry can be expressed as a composition of reflections.

Example

Describe the result of reflecting P first across line ℓ bold italic script l  and then across line m.

A composition of two reflections across intersecting lines is a rotation. The angle of rotation is twice the measure of the acute angle formed by the intersecting lines. P is rotated 100 ° 100 degrees  about C.

Exercises

26. Sketch and describe the result of reflecting E first across line ℓ script l  and then across line m.

    

Each figure is an isometry image of the figure below. Tell whether their orientations are the same or opposite. Then classify the isometry.

27. 
28. 
29. 
30. Δ T A M cap delta t eh m  has vertices T (0, 5), A(4, 1), and M(3, 6). Find the glide reflection image of Δ T A M cap delta t eh m  for the translation ( x , y ) → ( x − 4 , y ) open x comma y close rightwards arrow open x minus 4 comma y close  followed by reflection across the line y = − 2 . y equals negative 2 .

9-7 Tessellations

Quick Review

A tessellation is a repeating pattern of figures that completely covers a plane without gaps or overlaps. If the figures are polygons, the sum of the measures of the angles around any vertex in the tessellation is 360.

Every tessellation displays at least one type of symmetry: reflectional, rotational, translational, or glide reflectional symmetry.

Example

Does a regular decagon tessellate? Explain.

Use the Polygon Angle-Sum Theorem to find the measure, a, of each angle of a regular decagon.

a = 180 ( n − 2 ) n = 180 ( 10 − 2 ) 10 = 144 eh equals . fraction 180 . open , n minus 2 , close , over n end fraction . equals . fraction 180 . open , 10 minus 2 , close , over 10 end fraction . equals 144

The sum of the angle measures around one vertex of a tessellation must be 360. 144 is not a factor of 360, so a regular decagon does not tessellate.

Exercises

For each tessellation, (a) identify the repeating figure and the transformation used, and (b) list the types of symmetry the tessellation has.

Determine whether each figure tessellates. If so, draw a sketch. If not, explain.

33. a kite
34. a regular 14-gon
35. 
36. 

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