Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 1 AND
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1 Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 1 AND
2 Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 2
3 WHAT YOU WILL LEARN Transformational geometry, symmetry, and tessellations The Mobius Strip, Klein bottle, and maps Non-Euclidian geometry and fractal geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 3
4 Section 5 Transformational Geometry, Symmetry, and Tessellations Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 4
5 Definitions The act of moving a geometric figure from some starting position to some ending position without altering its shape or size is called a rigid motion (or transformation). Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 5
6 Reflection A reflection is a rigid motion that moves a geometric figure to a new position such that the figure in the new position is a mirror image of the figure in the starting position. In two dimensions, the figure and its mirror image are equidistant from a line called the reflection line or the axis of reflection. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 6
7 Construct the reflection of triangle ABC about the line l. A A C B l C 2 units 2 units B B l C A Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 7
8 Translation A translation (or glide) is a rigid motion that moves a geometric figure by sliding it along a straight line segment in the plane. The direction and length of the line segment completely determine the translation. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 8
9 Example Given the parallelogram and translation vector, v, construct the translated parallelogram. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 9
10 Example (continued) Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 10
11 Rotation A rotation is a rigid motion performed by rotating a geometric figure in the plane about a specific point, called the rotation point or the center of rotation. The angle through which the object is rotated is called the angle of rotation. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 11
12 Example Given the rectangle and rotation point, P, construct rectangles that result from rotations of 90º and 180º. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 12
13 Example (continued) 90º Rotation 180º Rotation Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 13
14 Glide Reflection A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 14
15 Example Construct a glide reflection of triangle ABC using translation vector v, and reflection line l. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 15
16 Symmetry A symmetry of a geometric figure is a rigid motion that moves a figure back onto itself. That is, the beginning position and ending position of the figure must be identical. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 16
17 Example Reflection about Line Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 17
18 Tessellations A tessellation (or tiling) is a pattern consisting of the repeated use of the same geometric figures to entirely cover a plane, leaving no gaps. The geometric figures used are called the tessellating shapes of the tessellation. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 18
19 Example The simplest tessellations use one single regular polygon. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 19
20 Example (continued) Other examples of tessellations: Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 20
21 Create Your Own Tessellation Copyright 2009 Pearson Education, Inc. Chapter 9 Section 5 - Slide 21
Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND
Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 2 WHAT YOU WILL LEARN Transformational geometry,
Geometric Transformations: Translation:
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