Chapter 12

 For each example, how would I get the first image to look like the second?

 What are these examples of?

 A transformation of a geometric figure is a change in its position, shape, or size.  Types of transformations: reflection (flip), translation (slide), rotation (turn), dilation (shrink or grow)  Preimage – original figure before the transformation  Image – resulting figure after the transformation

 An isometry is a transformation in which the preimage and image are congruent.  In other words, there is a change in position, but not shape or size.  A reflection is an isometry in which the orientation of the object and its image are opposites.

A reflection is an isometry in which the orientation of the object and its image are opposites.

 ABCD is an image of KLMN.  What is the image of angle L?  Which side corresponds to NK?  Sometimes images are named as A’B’C’D’ with the ‘ (prime) signifying the difference between the image and pre-image.

 ∆XYZ has vertices X(-2,3), Y(1,1), and Z(2,4). Draw ∆XYZ and its reflection image in the x-axis. Name using primes.

 ∆XYZ has vertices X(-2,3), Y(1,1), and Z(2,4). Draw ∆XYZ and its reflection image in the line x=3. Name using new letters.

 A translation is an isometry that maps all points of a figure the same distance in the same direction.  We describe translations using vectors

 Find the image of F under the translation.

 Find the vector that describes the translation H→I.

 Find the vector that describes the translation ∆ABC→ ∆A’B’C’.

 Draw the image of ∆ABC under the translation.

 To describe a rotation, you need three pieces of information: 1. center of rotation (a point on or off the figure)  ON  Off

2. angle of rotation (positive number, 360 max.) 3. direction of rotation (clockwise or counterclockwise)

Draw the image that results when ABC is rotated counterclockwise 270° around the origin.

 A composition of reflections in  two parallel lines is a translation.  two intersecting lines is a rotation.  A glide reflection is the composition of a glide (translation) and a reflection in a line parallel to the glide vector.

 A figure has symmetry if there is an isometry that maps the figure onto itself.  Three types of symmetry:  Line symmetry (a.k.a. reflectional symmetry)  Rotational symmetry – is its own image for some rotation that is less than or equal to 180°  Point symmetry – has rotational symmetry of exactly 180°

 What kind of symmetry does each figure have? (could be multiple types)

 A tessellation is a repeating pattern of figures that completely covers a plane, without gaps or overlaps.  All triangles and quadrilaterals tessellate.

A regular polygon will tessellate a plane if the interior angle measure will divide into 360 evenly.

 A dilation is a transformation whose preimage and image are similar.  It is generally not an isometry.

 Every dilation has a center and a scale factor.  The scale factor describes the size change from the original figure to the image.  The dilation is an enlargement if the scale factor n > 1.  It is a reduction if the scale factor 0 < n < 1.

 The green circle is a dilation of the blue circle. Describe the dilation.

 ∆ABC is a dilation of ∆DBC. Find the center and scale factor.

 The scale factor on a museum's floor plan is 1 : 200. The length and width on the drawing are 8 in. and 6 in. Find the actual dimensions in feet and inches.

 ∆XYZ has coordinates X(3,1), Y(2,-4), and Z (-2,0). Find the image for a dilation with center (0,0) and scale factor 2.5.