2. Adapted from Walch Education

3. 1.4.1: Describing Rigid Motions and Predicting the Effects 2 Rigid motions are transformations that don’t affect an object’s shape and size. This means that corresponding sides and corresponding angle measures are preserved. When angle measures and sides are preserved they are congruent, which means they have the same shape and size. The congruency symbol ( ) is used to show that two figures are congruent.

4. 1.4.1: Describing Rigid Motions and Predicting the Effects 3 The figure before the transformation is called the preimage. The figure after the transformation is the image. Corresponding sides are the sides of two figures that lie in the same position relative to the figure. In transformations, the corresponding sides are the preimage and image sides, so and are corresponding sides and so on.

5. 1.4.1: Describing Rigid Motions and Predicting the Effects 4 Corresponding angles are the angles of two figures that lie in the same position relative to the figure. In transformations, the corresponding vertices are the preimage and image vertices, so ∠ A and ∠ A′ are corresponding vertices and so on. Transformations that are rigid motions are translations, reflections, and rotations. Transformations that are not rigid motions are dilations, vertical stretches or compressions, and horizontal stretches or compressions.

6. 1.4.1: Describing Rigid Motions and Predicting the Effects 5 A translation is sometimes called a slide. In a translation, the figure is moved horizontally and/or vertically. The orientation of the figure remains the same. Connecting the corresponding vertices of the preimage and image will result in a set of parallel lines.

7. 1.4.1: Describing Rigid Motions and Predicting the Effects 6 Translating a Figure Given the Horizontal and Vertical Shift 1.Place your pencil on a vertex and count over horizontally the number of units the figure is to be translated. 2.Without lifting your pencil, count vertically the number of units the figure is to be translated. 3.Mark the image vertex on the coordinate plane. 4.Repeat this process for all vertices of the figure. 5.Connect the image vertices.

8. 1.4.1: Describing Rigid Motions and Predicting the Effects 7 A reflection creates a mirror image of the original figure over a reflection line. A reflection line can pass through the figure, be on the figure, or be outside the figure. Reflections are sometimes called flips. The orientation of the figure is changed in a reflection.

9. 1.4.1: Describing Rigid Motions and Predicting the Effects 8 In a reflection, the corresponding vertices of the preimage and image are equidistant from the line of reflection, meaning the distance from each vertex to the line of reflection is the same. The line of reflection is the perpendicular bisector of the segments that connect the corresponding vertices of the preimage and the image.

10. 1.4.1: Describing Rigid Motions and Predicting the Effects 9 Reflecting a Figure over a Given Reflection Line 1.Draw the reflection line on the same coordinate plane as the figure. 2.If the reflection line is vertical, count the number of horizontal units one vertex is from the line and count the same number of units on the opposite side of the line. Place the image vertex there. Repeat this process for all vertices. 3.If the reflection line is horizontal, count the number of vertical units one vertex is from the line and count the same number of units on the opposite side of the line. Place the image vertex there. Repeat this process for all vertices. (continued) 4.If the reflection line is diagonal, draw lines from each vertex that are perpendicular to the reflection line extending beyond the line of reflection. Copy each segment from the vertex to the line of reflection onto the perpendicular line on the other side of the reflection line and mark the image vertices. 5.Connect the image vertices.

11. 1.4.1: Describing Rigid Motions and Predicting the Effects 10 A rotation moves all points of a figure along a circular arc about a point. Rotations are sometimes called turns. In a rotation, the orientation is changed. The point of rotation can lie on, inside, or outside the figure, and is the fixed location that the object is turned around. The angle of rotation is the measure of the angle created by the preimage vertex to the point of rotation to the image vertex. All of these angles are congruent when a figure is rotated.

12. 1.4.1: Describing Rigid Motions and Predicting the Effects 11 Rotating a figure clockwise moves the figure in a circular arc about the point of rotation in the same direction that the hands move on a clock. Rotating a figure counterclockwise moves the figure in a circular arc about the point of rotation in the opposite direction that the hands move on a clock.

13. 1.4.1: Describing Rigid Motions and Predicting the Effects 12 Rotating a Figure Given a Point and Angle of Rotation 1.Draw a line from one vertex to the point of rotation. 2.Measure the angle of rotation using a protractor. 3.Draw a ray from the point of rotation extending outward that creates the angle of rotation. 4.Copy the segment connecting the point of rotation to the vertex (created in step 1) onto the ray created in step 3. 5.Mark the endpoint of the copied segment that is not the point of rotation with the letter of the corresponding vertex, followed by a prime mark (′ ). This is the first vertex of the rotated figure. 6.Repeat the process for each vertex of the figure. 7.Connect the vertices that have prime marks. This is the rotated figure.

14. 1.4.1: Describing Rigid Motions and Predicting the Effects 13 * Describe the transformation that has taken place in the diagram to the right.

15. 1.4.1: Describing Rigid Motions and Predicting the Effects 14 Examine the orientation of the figures to determine if the orientation has changed or stayed the same. ArmPreimage orientationImage orientation ShorterPointing upward from the corner of the figure with a negative slope at the end of the arm Pointing downward from the corner of the figure with a positive slope at the end of the arm LongerPointing to the left from the corner of the figure with a positive slope at the end of the arm Pointing to the left from the corner of the figure with a negative slope at the end of the arm

16. 1.4.1: Describing Rigid Motions and Predicting the Effects 15 * The orientation of the figures has changed. In the preimage, the outer right angle is in the bottom right-hand corner of the figure, with the shorter arm extending upward. In the image, the outer right angle is on the top right-hand side of the figure, with the shorter arm extending down. * PreimageImage

17. 1.4.1: Describing Rigid Motions and Predicting the Effects 16 Compare the slopes of the segments at the end of the longer arm. The slope of the segment at the end of the arm is positive in the preimage, but in the image the slope of the corresponding arm is negative. Preimage Image

18. 1.4.1: Describing Rigid Motions and Predicting the Effects 17 A similar reversal has occurred with the segment at the end of the shorter arm. In the preimage, the segment at the end of the shorter arm is negative, while in the image the slope is positive. Preimage Image

19. 1.4.1: Describing Rigid Motions and Predicting the Effects 18 Determine the transformation that has taken place. * Since the orientation has changed, the transformation is either a reflection or a rotation. Since the orientation of the image is the mirror image of the preimage, the transformation is a reflection. The figure has been flipped over a line.

20. 1.4.1: Describing Rigid Motions and Predicting the Effects 19 Determine the line of reflection. * Connect some of the corresponding vertices of the figure. Choose one of the segments you created and construct the perpendicular bisector of the segment. Verify that this is the perpendicular bisector for all segments joining the corresponding vertices. This is the line of reflection. * The line of reflection for this figure is y = –1, as shown on the next slide.

21. 1.4.1: Describing Rigid Motions and Predicting the Effects 20

22. 1.4.1: Describing Rigid Motions and Predicting the Effects 21 Rotate the given figure 45º counterclockwise about the origin.

23. Ms. Dambreville