Chapter 9 Geometry © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 9: Geometry 9.1 Points, Lines, Planes, and Angles 9.2 Curves, Polygons, and Circles 9.3 Perimeter, Area, and Circumference 9.4 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem 9.5Space Figures, Volume, and Surface Area 9.6Transformational Geometry 9.7 Non-Euclidean Geometry, Topology, and Networks 9.8 Chaos and Fractal Geometry

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 9-6 Transformational Geometry

© 2008 Pearson Addison-Wesley. All rights reserved Transformational Geometry Reflections Translations and Rotations Size Transformations

© 2008 Pearson Addison-Wesley. All rights reserved Transformational Geometry One branch of geometry, known as transformational geometry, investigates how one geometric figure can be transformed into another. In transformational geometry we are required to reflect, rotate, and change the size of the figures.

© 2008 Pearson Addison-Wesley. All rights reserved Reflections A m A’ Line m is perpendicular to the line segment AA’ and also bisects it. We call A’ the reflection image of the point A about the line m. Line m is called the line of reflection for points A and A’. The dashed line shows that the points are images of each other under this transformation.

© 2008 Pearson Addison-Wesley. All rights reserved Reflection The set of all reflection images of points of the original figure is called the reflection image of the figure. We call reflection about a line the reflection transformation. If a point A and its image, A’, under a certain transformation are the same point, then A is an invariant point of the transformation.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Reflection Image About Line m m m m

© 2008 Pearson Addison-Wesley. All rights reserved Reflection Three points that lie on the same line are called collinear. The reflection image of a line is also a line. Thus we say that reflection preserves collinearity.

© 2008 Pearson Addison-Wesley. All rights reserved Line of Symmetry The figure below is its own reflection image about the lines of reflection shown. In this case, the line of reflection is called a line of symmetry.

© 2008 Pearson Addison-Wesley. All rights reserved Composition We will use r m to represent a reflection about a line m, and let us use to represent a reflection about line m followed by a reflection about line n. We call the composition, or product, of the two reflections.

© 2008 Pearson Addison-Wesley. All rights reserved Translation m n The composition of two reflections about parallel lines is a translation.

© 2008 Pearson Addison-Wesley. All rights reserved Translations The distance between a point and its image under a translation is called the magnitude of the translation. A translation of magnitude 0 leaves every point of the plane unchanged and is called the identity translation. A translation of magnitude k, followed by a similar translation of magnitude k but of the opposite direction returns a point to its original position and these translations are called inverses of each other.

© 2008 Pearson Addison-Wesley. All rights reserved Rotations B m n The composition of two reflections about nonparallel lines is called a rotation. The point of intersection of these lines is called the center of rotation.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Rotation Find the image of a point P under a rotation transformation having center at a point Q and magnitude 90° clockwise. Solution 90° P Q P’

© 2008 Pearson Addison-Wesley. All rights reserved Point Reflection A rotation transformation having center Q and magnitude 180° clockwise is sometimes called a point reflection.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Point Reflection Find the point reflection image of the figure about point Q. Q Solution

© 2008 Pearson Addison-Wesley. All rights reserved Glide Reflection Let r m be a reflection about line m, and let T be a translation having nonzero magnitude and a direction parallel to m. The composition of T followed by r m is called a glide translation. m

© 2008 Pearson Addison-Wesley. All rights reserved Isometries Isometries are transformations in which the image of a figure has the same size and shape as the original figure.

© 2008 Pearson Addison-Wesley. All rights reserved Size Transformation A size transformation can have any positive real number k as magnitude. A size transformation having magnitude k > 1 is called a dilatation, or stretch; while a size transformation having magnitude k < 1 is called a contraction, or shrink.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Size Transformation M Apply a size transformation with center M and magnitude 1/3 to the triangle below. Solution

© 2008 Pearson Addison-Wesley. All rights reserved Example: Size Transformation M Apply a size transformation with center M and magnitude 1/3 to the triangle below. Solution

© 2008 Pearson Addison-Wesley. All rights reserved Summary of Transformations ReflectionTranslation Preserve collinearity ?Yes Preserve distance?Yes Identity?NoneMagnitude 0 Inverse?NoneSame magnitude; opposite direction Composition of n reflections? n = 1n = 2, parallel Isometry?Yes Invariant points?Line of reflectionNone

© 2008 Pearson Addison-Wesley. All rights reserved Summary of Transformations RotationGlide Reflection Preserve collinearity ?Yes Preserve distance?Yes Identity?Magnitude 360°None Inverse?Same center; magnitude (360 – x)° None Composition of n reflections? n = 2, nonparalleln = 3 Isometry?Yes Invariant points?Center of RotationNone

© 2008 Pearson Addison-Wesley. All rights reserved Summary of Transformations Size Transformation Preserve collinearity ?Yes Preserve distance?No Identity?Magnitude 1 Inverse?Same center; magnitude 1/k Composition of n reflections? No Isometry?No Invariant points?Center of transformation