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

2. 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.

3. 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.

4. 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.

5. 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.

6. Constructions of Translations To construct the translation image A′ of a point A with a compass and straightedge, construct a parallelogram MAA′N so that AA′ is in the same direction as MN. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. Example 13-2 (continued) The square ABCD and its image A′B′C′D′ are shown here. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

14. 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.

15. Rotations Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

16. 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.

17. Example 13-3 Find the image of ΔABC under the rotation shown with center O. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

18. 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.

19. 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.

20. Slopes of Perpendicular Lines Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

21. 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.

22. Example 13-4 Find the equation of line ℓ through point ( − 1, 2) and perpendicular to the line y = 3x + 5. The line y = 3x + 5 has slope 3, so the line ℓ has slope Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

23. Example 13-4 (continued) The equation of line ℓ is Copyright © 2013, 2010, and 2007, Pearson Education, Inc.