Transformations

1
Transformations

• Chapter 9

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Aim 9-1 How do we find translation images of
figures?

• A transformation of a geometric figure is a
  change in its position, shape or size.
• Example
• When you assemble a jigsaw puzzle

3
Identifying Isometries

• Does the transformation appear to be an isometry?
  Explain.
• No, this transformation involves a change in
  size. The sides of the preimage square and the
  sides of its image are not congruent.

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Identifying Isometries

• Does the transformation appear to be an isometry?
  Explain.

5

• A transformation maps a figure onto its image
  and may be described with arrow
• notation.
• Prime ( ) notation is sometimes used to identify
  the image points. In the diagram at the right ,
  K is the image of K (K K).

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Naming Images and Corresponding Parts

• In the diagram , EFGH is an image of EFGH.
• Name the image of lt F and lt H.
• ltF is the image of ltF.
• ltH is the image of ltH.
• List all pairs of corresponding sides.
• EF and EF FG and FG EH and EH
• GH and GH.

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Naming Images and Corresponding Parts

• In the diagram, NID SUP.
• Name all the images of ltI and point D.
• List all pairs of corresponding sides.

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Translations

• A translation or a slide is an isometry that maps
  all points of a figure the same distance in the
  same direction.

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Finding a Translation Image

• Find the image of ?XYZ under the translation (x,
  y) (x 2, y 5).

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Finding a Translation Image

• Use the rule, (x, y) (x 2, y 5). to find
  each vertex in the translated image.
• X(2, 1) translates to 2-2, 1-5 or X (0, -4).
• Y(3, 3) translates to 3-2, 3-5 or Y(1, -2).
• Z (-1,3) translates to -1-2, 3-5 or Z(-3, -2).

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Finding a Translation Image

• Find the image of ?XYZ for the translation (x, y)
  ( x- 4, y 1).

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Writing a Rule to Describe a Translation

• Write a rule to describe the translation PQRS
  PQRS.
• Use P(-1, -2) and its image P(-5, -1).
• X -5 (-1) -4
• Y -1 (-2) 1
• The rule is (x, y ) (x 4, y 1).

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• A composition of transformation is a combination
  of two or more transformations. In a composition,
  each transformation is performed on the image of
  the preceding transformation.

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SummaryAnswer in complete sentences.

• Write a rule to describe the translation ?GHI?
  ?G ?H ?I ?. Then find the image
• ?G ? ? H ? ? I? ?of ?G ? H ? I ? under the
  translation (x, y)?(x 8, y 6).

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Aim9-2 How do we find reflection images?

• A reflection (or flip) is an isometry in which a
  figure and it image have opposite orientations.
  The reflected image in a mirror appears
  backwards.

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Reflections

• You can use the two rules to reflect a figure
  across line r.
• If a point A is on line r, then the image of A is
  A itself (that is, A A).
• If a point B is not on line r, then r is the
  perpendicular bisector of BB.

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Reflections

• A point on the line of reflection is mapped onto
  itself.
• A point and its image are equidistant from the
  line of reflection.

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Reflections

• When you reflect a shape over the x-axis, the
  x-coordinate stays the same and the y-coordinate
  changes.
• (x, y)?(x, -y)
• When you reflect a shape over the y-axis, the
  x-coordinate changes and the y stays the same.
• (x, y) ?(-x, y)

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Finding Reflection Images

• If point P(2, -1) is reflected across the line y
  1, what are the coordinates of its reflection
  image?

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Finding Reflection Images

• What are the coordinates of the images of P if
  the reflection line is x 1?

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Finding Reflection Images

• Each point is reflected across the line
  indicated. Find the coordinates of the images.
• Q across x 1
• R across y -1
• S across the y-axis
• 4. U across x -3

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Drawing Reflection Images

• Given the points
• A(- 3, 4), B(0,1), C(2, 3), draw ?ABC and its
  reflection image across the x-axis.
• B. Reflect the ?ABC across the y-axis.

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Drawing Reflection Images

• Across X-axis

• Across the Y-axis

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Drawing Reflection Images

• Given the points A(- 3, 4), B(0,1), C(2, 3). Then
  reflect the image across the line x 3.

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SummaryAnswer in complete sentences.

• AB has endpoints A(2,-2) and B(5,3). Find the
  reflection image of the endpoints in the y-axis,
  the x-axis and the line x 8.

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Aim 9-3 How do we draw and identify rotations?

• To describe a rotation, you need to know the
  center of rotation ( a point), the angle of
  rotation ( a positive number of degrees), and
  whether the rotation is clockwise or
  counterclockwise.

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Identifying the Angle of Rotation

• Using a protractor measure the angle of rotation.

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Identifying the Angle of Rotation On the
Cartesian Plane

• Label the vertices.
• Draw a line from one vertex to the origin.
• Then draw another line from the image to the
  point of origin.
• Then identify the angle of rotation.
• The angle of rotation is ___.

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Rule for Rotating 180

• The rule for rotating 180 about the origin is
  (x, y) ? (-x, -y).

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Identifying the Rule for Rotating 90

• Find the angle of rotation for the hand-out 11.4
  problems 7 and 8.
• Part 2 On graph paper copy the first figure and
  label the vertices write the coordinates. Then,
  copy the rotated figure, label the vertices and
  write the coordinates.
• Can you come up with a rule for rotating an image
  90 counterclockwise about the origin?
• (x, y)? __

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• Repeat for problem 8. What is the rule for
  rotating a figure 90 clockwise?
• Then, complete problem 13-15 on 11.4.

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Rules for Rotating 90

• Motion Rule for 90 counterclockwise
• (x, y)?(-y, x)
• Motion Rule for 90clockwise
• (x, y) ? (y, -x)

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Regular Polygons

• A regular polygon has a center that is
  equidistant from its vertices. Segments that
  connect the center to the vertices divide the
  polygon into congruent triangles.
• You can use this fact to find rotation images of
  regular polygons.

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Identifying a Rotation Image

• Regular pentagon PENTA is divided into five
  congruent triangles.
• a. Name the image of E for 72 rotation about X.

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Identifying a Rotation Image

• b. Name the image of P for a 216 rotation about
  X.

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Identifying a Rotation Image

• Name the image of T for a 144 rotation about X.

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SummaryAnswer in complete sentences.

• Below, Figure A is rotated about the origin to
  produce Figure B. What is the angle of rotation?

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Aim 9-4How do we identify the type of symmetry
in a figure?

• A figure has symmetry if there is an isometery
  that maps the figure onto itself. If the isometry
  is the reflection of a plane figure, the figure
  has reflectional symmetry or line symmetry.

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Symmetry

• One half of the figure is a mirror image of its
  other half. Fold the figure along the line of
  symmetry and the halves match exactly.

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Identifying Lines of Symmetry

• Draw all lines of symmetry for a regular hexagon.

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Symmetry

• Draw a rectangle and all of its lines of
  symmetry.

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Symmetry

• A figure that has rotational symmetry is its own
  image for some rotation of 180 or less. A figure
  that has point symmetry has 180 rotational
  symmetry.
• A square has 90 and 180 rotational symmetry
  with the center of rotation at the center of the
  square. A square also has point symmetry.

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Identifying Rotational Symmetry

• Identify any rotational symmetry in the figure.

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Identifying Rotational Symmetry

• This figure has rotational symmetry, It will
  coincide with itself after being rotated 90 or
  180 in either direction.

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Identifying Rotational Symmetry

• This figure has rotational symmetry. It will
  coincide with itself after being rotated 60,
  120, or 180 in either direction.

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Identifying Rotational Symmetry

• This figure has no rotational symmetry. It does
  have horizontal line of symmetry.

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Identifying Rotational Symmetry

• Judging from appearance, tell whether each
  triangle has rotational symmetry. If so, give the
  angle of rotation.

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Identifying Rotational Symmetry

• The equilateral triangle has rotational symmetry.
  The angle of rotation is 120

• This isosceles triangle does not have rotational
  symmetry.

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Identifying Rotational Symmetry

• Judging from appearance, tell whether the figure
  at the right has rotational symmetry. If so, give
  the angle of rotation.
• Does the figure have point symmetry?

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• Three-dimensional objects have various types of
  symmetry about a line and reflectional symmetry
  in a plane.

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Symmetric Design

• Tell whether each object has rotational symmetry
  about a line and/ or reflectional symmetry in a
  plane.

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Symmetric Design

• Tell whether the umbrella has rotational symmetry
  about a line and/ or reflectional symmetry in a
  plane.

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SummaryAnswer in complete sentences.

• Create a license plate using two letters and 4
  digits that have the indicated symmetry.
• Rotational Symmetry
• Vertical Symmetry
• Horizontal Symmetry

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Aim 9-5 How do we locate dilation images?

• A dilation is a transformation who preimage and
  image are similar. A dilation is not an
  isometry.
• Every dilation has a center and a scale factor n,
  n gt0. The scale factor describes the size change
  from the original figure to the image.

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Finding a Dilation

• To find a dilation with center C and scale factor
  n, you can use the following two rules.
• The image C is itself (meaning CC)
• For any point R, R is on CR and CR nCR.

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• The dilation is an enlargement if the scale
  factor is gt 1.

• The dilation is a reduction if the scale factor
  is between 0 and 1.

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Finding a Scale Factor

• The blue triangle is a dilation image of the red
  triangle. Describe the dilation.
• The center is X. The image is larger than the
  preimage, so the dilation is an enlargement.

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Finding a Scale Factor

• The blue quadrilateral is a dilation image of the
  red quadrilateral. Describe the dilation.

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Graphing Dilation Images

• ?PZG has vertices P(2,0), Z(-1, ½), and G (1,
  -2).
• What are the coordinates of the image of P for a
  dilation with center (0,0) and scale factor 3?
• a) (5, 3) b) (6,0) c) (2/3, 0) d) (3, -6)

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Graphing Dilation Images

• Solution
• The scale factor is 3, so use the rule
• (x, y)?(3x, 3y).
• P(2,0) ?P(32, 30) or P(6, 0).
• The correct answer is B.
• What are the coordinates for G and Z?

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Graphing Dilations

• Find the image of ?PZG for a dilation with center
  (0,0) and scale factor ½. Draw the reduction.

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SummaryAnswer in complete sentences.

• An equilateral triangle has 4-in. sides. Describe
  its image for a dilation with scale factor 2.5
  Explain.
• True or False. Explain your answer.
• A dilation changes orientation.
• A dilation is an isometry.

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Aim 9-6 How do we use composition of
reflections?

• If two figures are congruent, there is a
  transformation that maps one onto the other.
• If no reflection is involved, then the figures
  are either translation or rotation images of each
  other.

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Identifying the Transformation

• The two figures are congruent. Is one figure a
  translation image of the other, a rotation image,
  or neither? Explain.

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Recognizing the Transformation

• The orientations of these congruent figures do
  not appear to be opposite, so one is a
  translation image or a rotation image of the
  other. Clearly, its not a translation image, so
  it must be rotation image.

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Recognizing the Transformation

• The two figures are congruent . Is one figure a
  translation image of the other, a rotation image,
  or neither. Explain.

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The two figures in each pair are congruent. Is
one figure a translation image of the other, a
rotation image or neither? Explain.
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• Any translation or rotation can be expressed as
  the composition of two reflections.
• Theorem 9-1
• A translation or rotation is composition of two
  reflections.

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Theorem 9-2

• A composition of reflections across two parallel
  lines is a translation.

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Theorem 9-3

• A composition of reflections across two
  intersecting lines is a rotation.

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Compositions of Reflections Across Parallel Lines

• Find the image R for a reflection across line l
  followed by a reflection across line m. Describe
  the resulting translation.

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Solution
R is translated the distance and direction shown
by the green arrow. The arrow is perpendicular to
lines l and m with length equal to twice the
distance from l and m
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Composition of Reflections in Intersecting lines

• Lines a and b intersect in point C and form an
  acute lt1 with measure 35. Find the image of R for
  a reflection across a line a and then a
  reflection across line b. Describe the resulting
  rotation.

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Find the image of each letter for a reflection
across line l and then a reflection across line
m. Describe the resulting translations or
rotation.
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Composition of Reflections in Intersecting lines

• Solution

R rotates clockwise through the angle shown by
the green arrow. The center of rotation is C and
the measure of the angle is twice the mlt1 or 70.
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Composition of Reflections in Intersecting lines

• Find the image of each letter for a reflection
  across line l and then a reflection across line
  m. Describe the resulting translations or
  rotation.

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Glide Reflections

• Two plane figures A and B can be congruent with
  opposite orientations. Reflect A and you get a
  figure A that has the same orientation as B. So,
  B is a translation or rotation image of A. By
  Theorem 9-1, two reflections map A to B. The net
  result is that three reflections map A to B.

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Glide Reflections

• Theorem 9-4
• Fundamental Theorem of Isometries
• In a plane, one or two congruent figures can be
  mapped onto the other by a composition of at most
  three reflections.

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Glide Reflections

• If two figures are congruent and have opposite
  orientations (but are not simply reflections of
  each other), then there is a slide and a
  reflection that will map one onto the other. A
  glide reflection is the composition of a glide
  (translation) and a reflection across a line
  parallel to the direction of translation.

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Finding a Glide Reflection Image

• Find the image of triangle TEX for a glide
  reflection where the translation is
• (x, y) ?(x, y-5) and the reflection line
• is x 0.

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Finding a Glide Reflection Image

• Solution

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Finding a Glide Reflection Image

• Use ?TEX.
• Find the image ?TEX under a glide reflection
  where the translation is (x, y)?(x 1, y) and
  the reflection line y -2.

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Finding a Glide Reflection Image

• Would the result of the last question be the same
  if you reflected ?TEX first, and then translated
  it? Explain.

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• You can map one of any two congruent figures onto
  the other by a single reflection, translation,
  rotation, or glide reflection . Thus, you are
  able to classify any isometry.

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Classifying Isometries

• Each figure is an isometry image of the figure at
  the left. Tell whether their orientation are the
  same or opposite. Then classify the isometry.

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• Classify the isometry.

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SummaryAnswer in complete sentences.

• Name four isometries. Then choose two, and
  explain which composition of transformations
  results in each.

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Aim 9-7 How do we identify transformations in
tessellations, and figures that will tessellate?

• A tessellation or tiling, is a repeating pattern
  of figures that completely covers a plane without
  gaps or overlaps.
• You can create tessellations with translations,
  rotations, and reflections. You can find
  tessellations in art, nature (ex. honeycomb), and
  everyday tiled floors.

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Identifying the Transformations in a Tessellations

• Identify a transformation and the repeating
  figures in this tessellation.

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Identifying the Transformations in a Tessellations

• Identify a transformation and the repeating
  figures in this tessellation.

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Determining Figures That Will Tessellate

• Because the figures in a tessellation do not
  overlap or leave gaps, the sum of the measures of
  the angles around any vertex must be 360. If the
  angles around a vertex are all congruent, then
  the measure of each angle must be a factor of
  360.

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Determining Figures That Will Tessellate

• Determine whether a regular 18-gon tessellates a
  plane.
• a 180 (n - 2 ) Use the formulas for the measure
• n of an angle of a
  regular polygon.
• Since 160 is
  not a factor of 360, the
  18-gon will not tessellate.

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Determining Figures That Will Tessellate

• Explain why you can tessellate a plane with an
  equilateral triangle.

95

• A figure does not have to be a regular polygon to
  tessellate.
• Theorem 9-6
• Every triangle tessellates.
• Explain why?

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• Theorem 9-7
• Every quadrilateral tessellates.
• Explain why?

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Identifying Symmetries in Tessellations

• The tessellations with regular hexagons at the
  right has reflectional symmetry in each of the
  blue lines. It has rotational symmetry centered
  at each of the red points.

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Identifying Symmetries in Tessellations

• The tessellation also has translational symmetry
  and
• A translation maps onto itself.

• Glide reflectional symmetry.
• A glide reflection maps onto itself.

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Identifying Symmetries in Tessellations

• List the symmetries in the tessellation.

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Identifying Symmetries in Tessellations

• Solution Rotational symmetry centered at each
  red point Translational symmetry (blue arrow)

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Identifying Symmetries in Tessellations

• List the symmetries in the tessellation.

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Creating Tessellations

• Draw a 1.5 inch square on a blank piece of paper
  and cut it out.
• Draw a curve joining two consecutive vertices.

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Creating Tessellations

• Cut along the curve you drew and slide the cutout
  piece to the opposite side of the square. Tape it
  in place.

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Creating Tessellations

• Repeat this process using the other two opposite
  sides of the square.

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Creating Tessellations

• Rotate the resulting figure. What does your
  imagination suggest it looks like?
• Is it a penguin wearing a hat or a knight on
  horseback? Could it be a dog with floppy ears?
  Draw the image on your figure.
• Create a tessellation using your figure.

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SummaryAnswer in complete sentences.

• A pure tessellation is a tessellation made up of
  congruent copies of one figure. Explain why there
  are three, and only three pure tessellations that
  use regular polygons.