Chapter 9 Geometry 2008 Pearson AddisonWesley All rights

Chapter 9: Geometry 9. 1 9. 2 9. 3 9. 4 9. 5 9.

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

Transformational Geometry • Reflections • Translations and Rotations • Size Transformations 4 © 2008

Transformational Geometry One branch of geometry, known as transformational geometry, investigates how one geometric

Reflections Line m is perpendicular to the line segment AA’ and also bisects it.

Reflection The set of all reflection images of points of the original figure is

Example: Reflection Image About Line m m 8 © 2008 Pearson Addison-Wesley. All rights

Reflection Three points that lie on the same line are called collinear. The reflection

Line of Symmetry The figure below is its own reflection image about the lines

Composition We will use rm to represent a reflection about a line m, and

Translation The composition of two reflections about parallel lines is a translation. m n

Translations The distance between a point and its image under a translation is called

Rotations The composition of two reflections about nonparallel lines is called a rotation. The

Example: Rotation Find the image of a point P under a rotation transformation having

Point Reflection A rotation transformation having center Q and magnitude 180° clockwise is sometimes

Example: Point Reflection Find the point reflection image of the figure about point Q.

Glide Reflection Let rm be a reflection about line m, and let T be

Isometries are transformations in which the image of a figure has the same size

Size Transformation A size transformation can have any positive real number k as magnitude.

Example: Size Transformation Apply a size transformation with center M and magnitude 1/3 to

Summary of Transformations Preserve collinearity ? Preserve distance? Identity? Inverse? Composition of n reflections?

Summary of Transformations Rotation Preserve collinearity ? Yes Preserve distance? Yes Identity? Magnitude 360°

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Chapter 9: Geometry 9. 1 9. 2 9. 3 9. 4 9. 5 9. 6 9. 7 9. 8 Points, Lines, Planes, and Angles Curves, Polygons, and Circles Perimeter, Area, and Circumference The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem Space Figures, Volume, and Surface Area Transformational Geometry Non-Euclidean Geometry, Topology, and Networks Chaos and Fractal Geometry 2 © 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. 5 © 2008 Pearson Addison-Wesley. All rights reserved

Reflections 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. A m A’ 6 © 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. 7 © 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. 9 © 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. 10 © 2008 Pearson Addison-Wesley. All rights reserved

Composition We will use rm 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. 11 © 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. 13 © 2008 Pearson Addison-Wesley. All rights reserved

Rotations 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. m n B 14 © 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 P Q 90° P’ 15 © 2008 Pearson Addison-Wesley. All rights reserved

Glide Reflection Let rm 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 rm is called a glide translation. m 18 © 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. 20 © 2008 Pearson Addison-Wesley. All rights reserved

Summary of Transformations Preserve collinearity ? Preserve distance? Identity? Inverse? Composition of n reflections? Isometry? Invariant points? Reflection Yes None Translation Yes Magnitude 0 Same magnitude; opposite direction n=1 n = 2, parallel Yes Line of reflection Yes None 23 © 2008 Pearson Addison-Wesley. All rights reserved

Summary of Transformations Rotation Preserve collinearity ? Yes Preserve distance? Yes Identity? Magnitude 360° Inverse? Same center; magnitude (360 – x)° Composition of n reflections? Isometry? Invariant points? Glide Reflection Yes None n = 2, nonparallel n=3 Yes Center of Rotation Yes None 24 © 2008 Pearson Addison-Wesley. All rights reserved

Summary of Transformations Preserve collinearity ? Preserve distance? Identity? Inverse? Composition of n reflections? Isometry? Invariant points? Size Transformation Yes No Magnitude 1 Same center; magnitude 1/k No No Center of transformation 25 © 2008 Pearson Addison-Wesley. All rights reserved