7.1 Rigid Motion in a Plane

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7.1 Rigid Motion in a Plane
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Identifying Transformations

• Figures in a plane can be
• Reflected
• Rotated
• Translated
• To produce new figures. The new figures is
  called the IMAGE.
• The original figures is called the PREIMAGE.
• The operation that MAPS, or moves the preimage
  onto the image is called a TRANSFORMATION.

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What will you learn?

• Three basic transformations
• Reflections
• Rotations
• Translations
• And combinations of the three.
• For each of the three transformations on the next
  slide, the blue figure is the preimage and the
  red figure is the image. We will use this color
  convention throughout the rest of the book.

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Copy this down
Rotation about a point
Reflection in a line
Translation
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Some facts

• Some transformations involve labels.
• When you name an image, take the corresponding
  point of the preimage and add a prime symbol.
• For instance, if the preimage is A, then the
  image is A, read as A prime.

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Example 1 Naming transformations

• Use the graph of the transformation at the right.
• Name and describe the transformation.
• Name the coordinates of the vertices of the
  image.
• Is ?ABC congruent to its image?

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Example 1 Naming transformations

• Name and describe the transformation.
• The transformation is a reflection in the y-axis.
  You can imagine that the image was obtained by
  flipping ?ABC over the y-axis.

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Example 1 Naming transformations

• Name the coordinates of the vertices of the
  image.
• The cordinates of the vertices of the image,
  ?ABC, are A(4,1), B(3,5), and C(1,1).

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Example 1 Naming transformations

• Is ?ABC congruent to its image?
• Yes ?ABC is congruent to its image ?ABC. One
  way to show this would be to use the DISTANCE
  FORMULA to find the lengths of the sides of both
  triangles. Then use the SSS Congruence Postulate

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ISOMETRY

• An ISOMETRY is a transformation the preserves
  lengths.
• Isometries also preserve angle measures, parallel
  lines, and distances between points.
• Transformations that are isometries are called
  RIGID TRANSFORMATIONS.

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Ex. 2 Identifying Isometries

• Which of the following appear to be isometries?
• This transformation appears to be an isometry.
  The blue parallelogram is reflected in a line to
  produce a congruent red parallelogram.

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Ex. 2 Identifying Isometries

• Which of the following appear to be isometries?
• This transformation is not an ISOMETRY because
  the image is not congruent to the preimage

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Ex. 2 Identifying Isometries

• Which of the following appear to be isometries?
• This transformation appears to be an isometry.
  The blue parallelogram is rotated about a point
  to produce a congruent red parallelogram.

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Mappings

• You can describe the transformation in the
  diagram by writing ?ABC is mapped onto ?DEF.
  You can also use arrow notation as follows
• ?ABC ? ?DEF
• The order in which the vertices are listed
  specifies the correspondence. Either of the
  descriptions implies that
• A ? D, B ? E, and C ? F.

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Ex. 3 Preserving Length and Angle Measures

• In the diagram ?PQR is mapped onto ?XYZ.
• The mapping is a rotation.
• Given that ?PQR ? ?XYZ is an isometry, find the
  length of XY and the measure of ?Z.

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Ex. 3 Preserving Length and Angle Measures

• SOLUTION
• The statement ?PQR is mapped onto ?XYZ implies
  that P ? X, Q ? Y, and R ? Z. Because the
  transformation is an isometry, the two triangles
  are congruent.
• ?So, XY PQ 3 and m?Z m?R 35.

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Using Transformations in Real Life

• Look at sample problems on page 398.