Properties of Rotations 8.10.A. Sketch the image of the rotation. Label the images of points A, B, and C as A, B, and C.

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1 ? LESSON 1.3 ESSENTIL QUESTION Properties of Rotations How do ou describe the properties of orientation and congruence of rotations? Two-dimensional shapes Generalize the properties of orientation and congruence of rotations of twodimensional shapes on a coordinate plane. EXPLORE TIVITY Eploring Rotations rotation is a transformation that turns a figure around a given point called the center of rotation. The image has the same size and shape as the preimage. The triangle shown on the grid is the preimage. You will use the origin as the center of rotation. Houghton Mifflin Harcourt Publishing ompan Image redits: IKO/Fotolia E Trace triangle onto a piece of paper. ut out our traced triangle. Rotate our triangle 90 counterclockwise about the origin. The side of the triangle that lies along the -ais should now lie along the -ais. Sketch the image of the rotation. Label the images of points,, and as,, and. escribe the motion modeled b the rotation. Rotate about the origin. degrees heck that the motion ou described in is the same motion that maps point onto, point onto, and point onto. Reflect 1. ommunicate Mathematical Ideas How are the size and the orientation of the triangle affected b the rotation? - -. Rotate triangle 90 clockwise about the origin. Sketch the result on the coordinate grid above. Label the image vertices,, and. Lesson

2 EXPLORE TIVITY Properties of Rotations Use trapezoid TRP to investigate the properties of rotations. Trace the trapezoid onto a piece of paper. Include the portion of the - and -aes bordering the third quadrant. ut out our tracing. 6 Place our trapezoid and aes on top of those in the figure. Then use the aes to help rotate our trapezoid 180 counterclockwise about the origin. Sketch the image of the rotation of our trapezoid in this new location. Label the vertices of the image T, R,, and P. -6 O 6 R Use a ruler to measure the sides of trapezoid TRP in centimeters. -6 T P TR = R = P = TP = Use a ruler to measure the sides of trapezoid T R P in centimeters. T R = R = P = T P = E What do ou notice about the lengths of corresponding sides of the two figures? F G H I Use a protractor to measure the angles of trapezoid TRP. m T = m R = m = m P = Use a protractor to measure the angles of trapezoid T R P. m T = m R = m = m P = What do ou notice about the measures of corresponding angles of the two figures? Which sides of trapezoid TRP are parallel? Which sides of trapezoid T R P are parallel? What do ou notice? Houghton Mifflin Harcourt Publishing ompan 346 Unit

3 Reflect 3. Make a onjecture Use our results from E, H, and I to make a conjecture about rotations. 4. Place our tracing back in its original position. Then perform a 180 clockwise rotation about the origin. ompare the result. Graphing Rotations To rotate a figure in the coordinate plane, rotate each of its vertices. Then connect the vertices to form the image. EXMPLE Math On the Spot m.hrw.com The figure shows triangle. Graph the image of triangle after a rotation of 90 clockwise. STEP 1 Rotate the figure clockwise from the -ais to the -ais. Point will still be at (0, 0). - - nimated Math m.hrw.com Houghton Mifflin Harcourt Publishing ompan STEP Point is units to the left of the -ais, so point is units above the -ais. Point is units to the right of the -ais, so point is units below the -ais. onnect,, and to form the image triangle. Reflect. Is the image congruent to the preimage? How do ou know? Math Talk Mathematical Processes How is the orientation of the triangle affected b the rotation? - - Lesson

4 Personal Math Trainer Online ssessment and Intervention YOUR TURN Graph the image of quadrilateral after each rotation m.hrw.com clockwise 8. Find the coordinates of Point after a 90 counterclockwise rotation followed b a 180 rotation. - - Guided Practice 1. Vocabular rotation is a transformation that turns a figure around a given called the center of rotation. Siobhan rotates a right triangle 90 counterclockwise about the origin.. How does the orientation of the image of the triangle compare with the orientation of the preimage? (Eplore ctivit 1) 3. Is the image of the triangle congruent to the preimage? (Eplore ctivit )? raw the image of the figure after the given rotation about the origin. (Eample 1) counterclockwise E - O F - O G -4-4 ESSENTIL QUESTION HEK-IN 6. What are the properties of rotations? Houghton Mifflin Harcourt Publishing ompan 348 Unit

5 Name lass ate 1.3 Independent Practice m.hrw.com Personal Math Trainer Online ssessment and Intervention 7. The figure shows triangle and a rotation of the triangle about the origin. a. How would ou describe the rotation? ' ' - O ' b. What are the coordinates of the image?,, - 8. The graph shows a figure and its image after a transformation. a. How would ou describe this as a rotation? - - b. an ou describe this as a transformation other than a rotation? Eplain. Houghton Mifflin Harcourt Publishing ompan 9. What tpe of rotation will preserve the orientation of the H-shaped figure in the grid? 10. point with coordinates (-, -3) is rotated 90 clockwise about the origin. What are the coordinates of its image? omplete the table with rotations of 180 or less. Include the direction of rotation for rotations of less than O -3 3 Shape in quadrant Image in quadrant Rotation 11. I IV 1. III I 13. IV III Lesson

6 raw the image of the figure after the given rotation about the origin counterclockwise - - O Is there a rotation for which the orientation of the image is alwas the same as that of the preimage? If so, what? FOUS ON HIGHER ORER THINKING Work rea 17. Problem Solving Lucas is plaing a game where he has to rotate a figure for it to fit in an open space. Ever time he clicks a button, the figure rotates 90 degrees clockwise. How man times does he need to click the button so that each figure returns to its original orientation? Figure Figure Figure 18. Make a onjecture Triangle is reflected across the -ais to form the image. Triangle is then reflected across the -ais to form the image. What tpe of rotation can be used to describe the relationship between triangle and triangle? 19. ommunicate Mathematical Ideas Point is on the -ais. escribe all possible locations of image for rotations of 90, 180, and 70. Include the origin as a possible location for. Houghton Mifflin Harcourt Publishing ompan 30 Unit