E9 Students are expected to make generalizations about the properties of translations and reflections and apply these properties.

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E9 Students are expected to make generalizations
about the properties of translations and
reflections and apply these properties.
E10 Students are expected to explore rotations
of one-quarter, one-half, and three-quarter turns
using a variety of centres.
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What is Transformational Geometry?

• Transformational geometry is the study of
• transformations.

What is a transformation?
A transformation is a change in size or position
of a geometric figure.
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Types of Transformations

• Reflections or what are called flips in the
  younger grades.
• Translations or what are called slides in the
  younger grades.
• Rotations or what are called turns in the
  earlier grades.
• Dilatations that are introduced in Grade Six.

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What is a Reflection in Transformational Geometry?

• A reflection is the figure formed by flipping or
  reflecting a geometric figure about a line to get
  a mirror or reflection image.
• The original figure is referred to as the object
  while the figure created by the reflection of
  that object is called the reflection image.
• Now lets create some reflection images and try
  to discover the properties of a reflection.

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Investigating Reflections

• Draw a square on white paper. Draw a mirror line
  or a line of reflection. Place your mira on this
  line, and then draw the reflected image.
• Draw another shape. Draw a mirror line or a line
  of reflection. Place your mira on this line, and
  then draw the reflected image.
• Compare your original drawings (objects) with the
  reflected images by tracing the object and then
  fitting it on its reflected image. What can you
  say about the object and its reflected image?
• You should have concluded that the object and its
  reflected image are congruent.

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Investigating Reflections

• Draw a trapezoid and label its vertices with A,
  B, C, and D.
• Next, label the corresponding vertices of the
  reflected image with A, B, C, D.
• Name both shapes clockwise starting at A and A.
  What do you notice?
• You should have noticed that the object and its
  image are of opposite orientation. See the next
  slide to review this.

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What do you notice in the diagram below?
A
A
D
D
B
B
C
C
The object and its reflected image are of
opposite orientation.
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Investigating Reflections

• Now examine the trapezoid and its reflected image
  again.
• Make line segments by joining the corresponding
  vertices (A to A, etc.). Examine the angles made
  by the mirror lines with these segments. What can
  you conclude?
• You should have concluded that the mirror lines
  are perpendicular to the line segments joining
  the corresponding image points.

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Investigating Reflections

• Now examine the trapezoid and its reflected image
  again.
• Measure the distance from one vertex in the
  original trapezoid to the mirror line. Then
  measure the distance of the corresponding vertex
  in its image to the mirror line. Do this for the
  other pairs of points. What do you notice?
• You should have noticed that the corresponding
  points are equidistant from the mirror lines.
• You should now notice that the mirror line is the
  perpendicular bisector of all segments joining
  corresponding points.

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Properties of Reflections A Review

• The image is congruent to the object.
• The orientation of the image is reversed. If
  triangle ABC is read clockwise, then triangle
  ABC is read counterclockwise.
• If corresponding points on the object and image
  are joined, each line segment makes a right angle
  with the reflection or mirror line.
  (perpendicular). In other words, the mirror lines
  are perpendicular to the line segments joining
  the corresponding image points.
• Any point and its image are the same distance
  from the reflection line. The mirror line is the
  perpendicular bisector of all segments joining
  corresponding points.

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Lets Practice!

• Use the properties of reflections to draw
  reflected images from objects presented to you by
  the teacher.

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What is a Translation in Transformational
Geometry?

• A translation is a movement of a geometric figure
  to a new position by sliding it along a plane.
• The original figure is referred to as the object
  while the figure created by the translation or
  slide of that object is called the translated
  image or sometimes just the image.
• Now lets create some translated images and try
  to discover the properties of a translation.

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Investigating Translations

• Draw a square on white paper. Trace around the
  square and then take this tracing and translate
  it on the plane or grid paper according to the
  slide rule given to you by the teacher.
• Compare the size of the object with its
  translated image. What can you say about the
  object and its image?
• You should have concluded that the object and its
  translated image are congruent.

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Investigating Translations

• Draw a trapezoid and label its vertices with A,
  B, C, and D.
• Next, label the corresponding vertices of the
  translated image with A, B, C, D.
• Name both shapes clockwise starting at A and A.
  What do you notice?
• You should have noticed that the shape and its
  image are of the same orientation. See the next
  slide to review this.

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What do you notice in the diagram below?
A
D
A
B
D
C
B
C
The object and its translated image are of the
same orientation.
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Investigating Translations

• Now examine the trapezoid and its translated
  image again.
• What do you notice about line segment AB and its
  corresponding line segment AB? What about line
  segment DC and line segment DC? Look at the
  other corresponding line segments to see if you
  notice the same property.
• You should have noticed that the corresponding
  sides of the object and its translated image are
  parallel to one another.

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Investigating Translations

• Now examine the trapezoid and its translated
  image again.
• Measure the distance from one vertex in the
  original trapezoid to its corresponding vertex in
  the image. Next, measure the distance from a
  second vertex in the object to its corresponding
  vertex in the image. Do the same for the two
  other sets of corresponding vertices. What do you
  notice about these measurements?
• You should have noticed that all of the line
  segments joining corresponding points are equal
  in length and parallel to one another.

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Properties of Translations A Review

• The image is congruent to the object.
• The image has the same orientation as the object.
• The line segments of the image are parallel to
  the corresponding line segments of the object.
• The line segment joining the vertices of the
  object to the corresponding image vertices are
  equal in length and parallel.

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Lets Practice!

• Use the properties of translations to draw
  translated images from objects presented to you
  by the teacher.

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What is a Rotation in Transformational Geometry?

• A rotation is the movement that results when a
  geometric figure is turned about a fixed point.
• The original figure is referred to as the object
  while the figure created by the reflection of
  that object is called the rotated image.
• A rotation can be created by rotating an object
  about its centre point or about a another fixed
  point on the figure even a point somewhere
  outside of the figure on the plane.

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Investigating Quarter-Turn Rotations

• Draw a square on white paper. Mark its vertices
  with the points A, B, C, and D. Next, mark the
  squares centre point with the point E. Hold the
  square at this centre point, and rotate it 90
  degrees.
• This is called a quarter turn of the square about
  its center point. Save your square for a later
  activity.
• Now use a different figure such as a trapezoid.
  Label its vertices A, B, C, and D and its centre
  point E. Now make a quarter turn about the
  trapezoids centre. Compare the object with its
  rotated image. What can you say about the size of
  the object compared to its image?
• You should have concluded that the object and its
  rotated image are congruent. Save your trapezoid.

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Investigating Half-Turn Rotations

• Use your labeled square from the previous set of
  activities. Hold the square at this centre
  point, and this time rotate it 180 degrees.
• This is called a half turn of the square about
  its center point.
• Now use your trapezoid from the previous set of
  activities. This time use it to make a half turn
  about the trapezoids centre. Compare the object
  with its rotated image. What can you say about
  the size of the object compared to its image?
• You should have concluded that the object and its
  rotated image are congruent.

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Investigating Quarter-Turn Rotations

• Examine your object (trapezoid) and its image
  again.
• What labels should you place on the rotated
  image?
• Name both shapes clockwise starting at A and A.
  What do you notice?
• You should have noticed that the shape and its
  rotated image are of the same orientation.

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Investigating Half-Turn Rotations

• Examine your object (trapezoid) and its image
  again.
• What labels should you place on the rotated
  image?
• Name both shapes clockwise starting at A and A.
  What do you notice?
• You should have noticed that the shape and its
  rotated image are of the same orientation.

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Investigating Quarter-Turn Rotations

• Examine your object (trapezoid) and its image
  again.
• Measure the distance between any point on the
  object to the centre of the object. Now measure
  the distance between that point on the image to
  the centre of the image. What do you notice?
• You should have noticed that any point and its
  image are the same distance from the rotation
  centre.

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Investigating Half-Turn Rotations

• Examine your object (trapezoid) and its image
  again.
• Measure the distance between any point on the
  object to the centre of the object. Now measure
  the distance between that point on the image to
  the centre of the image. What do you notice?
• You should have noticed that any point and its
  image are the same distance from the rotation
  centre.

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Investigating Quarter-Turn Rotations

• Examine your object (trapezoid) and its image
  again.
• Place a tracing of the image on top of the
  object. Draw a line segment from any point on
  the object to the turn centre of the object. Draw
  a similar line segment from a corresponding point
  on the image to the centre of the image. What do
  you notice about the angle formed by these two
  segments?
• You should have noticed corresponding points
  drawn to the turn centre will form the angle of
  turn, in this case a 90-degree angle.

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Investigating Half-Turn Rotations

• Examine your object (trapezoid) and its image
  again.
• Place a tracing of the image on top of the
  object. Draw a line segment from any point on
  the object to the turn centre of the object. Draw
  a similar line segment from a corresponding point
  on the image to the centre of the image. What do
  you notice about the angle formed by these two
  segments?
• You should have noticed corresponding points
  drawn to the turn centre will form the angle of
  turn, in this case a 180-degree angle.

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Investigating Half-Turn Rotations

• Now examine the trapezoid and its rotated image
  again.
• What do you notice about line segment AB and its
  corresponding line segment AB? What about line
  segment DC and line segment DC? Look at the
  other corresponding line segments to see if you
  notice the same property.
• You should have noticed that the corresponding
  sides of the object and its half-turn rotated
  image are parallel to one another.

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Properties of Quarter-Turn Rotations A Review

• The image is congruent to the object.
• The image has the same orientation as the object.
• Any point and its image are the same distance
  from the rotation centre.
• Corresponding points drawn to the turn centre
  will form the angle of turn.

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Properties of Half-Turn Rotations A Review

• The image is congruent to the object.
• The image has the same orientation as the object.
• Any point and its image are the same distance
  from the rotation centre.
• Corresponding points drawn to the turn centre
  will form the angle of turn.
• Pairs of corresponding line segments are
  parallel.

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Investigating More about Rotations

• Now make three-quarter rotations of one of your
  figures. What properties do you notice?
• Now investigate with the teacher to see if the
  properties of quarter, half, and three quarter
  rotations hold turn if the turn point is not the
  centre of the object but another point on the
  object or a point that is not located on the
  object but is somewhere outside of it.

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Well, students, this is about all I can tell you
right now about reflections, translations, and
rotations. Stay tuned for Grade Six.