13 Chapter Congruence and Similarity with Transformations

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Presentation transcript:

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

13-2 Reflections and Glide Reflections Constructing a Reflection by Using Tracing Paper Constructing a Reflection on Dot Paper or a Geoboard Reflections in a Coordinate System Glide Reflections Congruence via Isometries Light Reflecting from a Surface Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Reflections Another isometry is a reflection, or flip. One example of a reflection often encountered in our daily lives is a mirror image. A reflection in a line ℓ is a transformation of a plane that pairs each point P of the plane with a point P′ in such a way that ℓ is the perpendicular bisector of PP′, as long as P is not on ℓ. If P is on ℓ, then P = P′. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Reflections We can obtain reflections in a line in various ways.Folding the paper along the reflecting line and drawing the image gives the mirror image, or image, of the half tree. Another way to simulate a reflection in a line involves using a Mira. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Constructing a Reflection by Using Tracing Paper Trace the original figure, the reflecting line, and a point on the reflecting line, which we use as a reference point. Flip the tracing paper over to perform the reflection, and align the reflecting line and the reference point. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Constructing a Reflection on Dot Paper or a Geoboard The image AB is reflected about line m. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 13-6 Find the image of ΔABC under a reflection in line m. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Reflections in a Coordinate System For some reflecting lines, like the x-axis and y-axis and the line y = x, it is quite easy to find the coordinates of the image, given the coordinates of the point. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Reflections in a Coordinate System The image of A(1, 4) is A′(4, 1). The image of B(−3, 0) is B′ (0, −3). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Light Reflecting from a Surface Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Light Reflecting from a Surface When a ray of light bounces off a mirror, the angle of incidence (the angle formed by the incoming rays and a line perpendicular to the mirror), is congruent to the angle of reflection (the angle between the reflected ray and the line perpendicular to the mirror). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Glide Reflections Another basic isometry, a glide reflection, is a transformation consisting of a translation followed by a reflection in a line parallel to the slide arrow. Translation image of C1 Original C2 C1 m Reflection image of C2 Glide reflection image of C1 C3 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Congruence via Isometries It is possible to define two figures as congruent if, and only if, one is an image of the other under an isometry or a composition of isometries. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 13-7 ABCD is a rectangle. Describe a sequence of isometries to show: a. ΔADC  ΔCBA A half-turn of ΔADC with center E is one transformation. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 13-7 (continued) b. ΔADC  ΔBCD A reflection in a line passing through E and parallel to AD is one transformation. c. ΔADC  ΔDAB A reflection of ΔADC in a line passing through E and parallel to DC is one transformation. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.