Chapter Congruence and Similarity with Transformations 13 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

13-3 Dilations Size Transformations Applications of Size Transformations Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Dilations Isometries preserve distance; so an image under isometry is congruent to the original image. A different type of transformation is created when a slide is projected on a screen. The objects on the screen are enlarged from their original size on the screen by the same factor. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Scale factor of 20 Dilations Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Dilations Each side of ΔA′B′C′ is twice as long as the corresponding sides of ΔABC. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Dilations A dilation from the plane to the plane with center O and scale factor r (r > 0) is a transformation that assigns to each point A in the plane the point A′ such that O, A, and A′ are collinear and OA′ = rOA and such that O is not between A and A′. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Dilations It is possible to define a size transformation when the scale factor is negative in the same way as in the preceding definition for positive scale factor, except that O must be between A and A′. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 13-8 Find the image of the quadrilateral ABCD under a size transformation with center O and scale factor Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

A dilation with center O and scale factor r (r > 0) has the following properties: Dilations  The image of a line segment is a line segment parallel to the original segment and r times as long.  The image of an angle is an angle congruent to the original angle. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 13-9 Show that ΔABC is the image of ΔADE under a size transformation. Identify the center of the size transformation and the scale factor., so A is the center of the size transformation and the scale factor is 2. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example Show that ΔABC is the image of ΔAPQ under a succession of isometries with a size transformation. Transform ΔAPQ by a half-turn in A to obtain ΔAP′Q′. C is the image of Q′ under a size transformation with center at A and scale factor 2. B is the image of P′ and A is the image of itself under this transformation. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example (continued) Therefore ΔABC can be obtained from ΔAPQ by first finding the image of ΔAPQ under a half-turn in A and then applying a size transformation with center A and scale factor 2 to that image. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Similar Figures Two figures are similar if it is possible to transform one onto the other by a sequence of isometries followed by a size transformation. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Applications of Size Transformations One way to make an object appear three- dimensional is to use a perspective drawing. For example, to make a letter appear three- dimensional we can use a size transformation with an appropriate center O and a scale factor with selected images to create a three dimensional effect for the letter C. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Applications of Size Transformations Copyright © 2013, 2010, and 2007, Pearson Education, Inc.