Chapter Congruence and Similarity with Transformations 13 Copyright 2013, 2010, and 2007, Pearson Education, Inc. 13-1 Translations and Rotations Translations Constructions of Translations Coordinate Representation of Translations Rotations Slopes of Perpendicular Lines Copyright 2013, 2010, and 2007, Pearson Education, Inc. Isometries Any rigid motion that preserves length or distance is an isometry. A translation is one type of isometry. Any function from a plane to itself that is a one-to- one correspondence is a transformation of the plane. Because transformations are functions, they have properties of functions. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Properties of Translations A translation is a motion of a plane that moves every point of the plane a specified distance in a specified direction along a straight line. A figure and its image are congruent. The image of a line is a line parallel to it. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Constructions of Translations To construct the image of an object under a translation, we first need to know how to construct the image A of a single point A. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Constructions of Translations To construct the translation image A of a point A with a compass and straightedge, construct a parallelogram MAAN so that AA is in the same direction as MN. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-1 Find the image of AB under the translation from X to X pictured on the dot paper. The image of each point on the dot paper can be obtained by first shifting it 2 units down and then 2 units to the right. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Point (x, y) Image Point (x + 5, y 2) A( 2, 6)A(3, 4) B(0, 8)B(5, 6) C(2, 3)C(7, 1) Coordinate Representation of Translations Copyright 2013, 2010, and 2007, Pearson Education, Inc. Translation in a Coordinate System A translation is a function from the plane to the plane such that to every point (x, y) corresponds the point (x + a, y + b), where a and b are real numbers. This can be written symbolically as (x, y) (x + a, y + b) where denotes moves to. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-2 Find the coordinates of the image of the vertices of quadrilateral ABCD under the translations in parts (a) through (c). a. (x, y) (x 2, y + 4) A(0, 0) A( 2, 4) B(2, 2) B(0, 6) C(4, 0) C(2, 4) D(2, 2) D(0, 2) Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-2 (continued) b.A translation determined by the slide arrow from A(0, 0) to A( 2, 4). This is the same translation as in part (a). Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-2 (continued) c.A translation determined by the slide arrow from S(4, 3) to S(2, 1). The translation from S(4, 3) to S(2, 1) moves an x-coordinate 2 units and a y-coordinate 4 units. Thus, every point (x, y) moves to a point with coordinates (x 2, y + 4). This is the same translation as in part (a). Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-2 (continued) The square ABCD and its image ABCD are shown here. Copyright 2013, 2010, and 2007, Pearson Education, Inc. A rotation is a transformation of the plane determined by holding one point, the center, fixed and rotating the plane about this point by a certain amount in a certain direction (a certain number of degrees either clockwise or counterclockwise). Rotations Copyright 2013, 2010, and 2007, Pearson Education, Inc. Rotations Copyright 2013, 2010, and 2007, Pearson Education, Inc. Isometries It can be shown that under any isometry, the image of a line is a line, the image of a circle is a circle, and the images of parallel lines are parallel lines. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-3 Find the image of ABC under the rotation shown with center O. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Special Rotations A rotation of 360 about a point moves any point and hence any figure onto itself. Such a transformation is an identity transformation. Any point may be the center of such a rotation. A rotation of 180 about a point is a half-turn. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Slopes of Perpendicular Lines We can use transformations to investigate various mathematical relationships, including the slopes between two non-vertical perpendicular lines. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Slopes of Perpendicular Lines Copyright 2013, 2010, and 2007, Pearson Education, Inc. Slopes of Perpendicular Lines Two lines, neither of which is vertical, are perpendicular if, and only if, their slopes m 1 and m 2 satisfy the condition m 1 m 2 = 1. Every vertical line has no slope but is perpendicular to a line with slope 0. Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-4 Find the equation of line through point ( 1, 2) and perpendicular to the line y = 3x + 5. The line y = 3x + 5 has slope 3, so the line has slope Copyright 2013, 2010, and 2007, Pearson Education, Inc. Example 13-4 (continued) The equation of line is Copyright 2013, 2010, and 2007, Pearson Education, Inc.