17.1 Translations. Exploring Translations. Explore

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1 Name lass Date 17.1 Translations Essential Question: How do ou draw the image of a figure under a translation? Eplore Eploring Translations translation slides all points of a figure the same distance in the same direction. Resource Locker You can use tracing paper to model translating a triangle. First, draw a triangle on lined paper. Label the vertices,, and. Then draw a line segment XY. n eample of what our drawing ma look like is shown. Use tracing paper to draw a cop of triangle. Then cop _ XY so that the point X is on top of point. Label the point made from Y as '. X Y X Y Houghton Mifflin Harcourt Publishing ompan Using the same piece of tracing paper, place ' on and draw a cop of. Label the corresponding vertices ' and '. n eample of what our drawing ma look like is shown. ' D Use a ruler to draw line segments from each verte of the preimage to the corresponding verte on the new image. ' X Y X Y Module Lesson 1

2 E Measure the distances ', ', ', and XY. Describe how ', ', and ' compare to the length XY. Reflect 1. re ', ', and ' parallel, perpendicular, or neither? Describe how ou can check that our answer is reasonable.. How does the angle relate to the angle '''? Eplain. Eplain 1 Translating Figures Using Vectors vector is a quantit that has both direction and magnitude. The initial point of a vector is the starting point. The terminal point of a vector is the ending point. The vector shown ma be named EF or v. E Initial point F Terminal point Translation It is convenient to describe translations using vectors. translation is a transformation along a vector such that the segment joining a point and its image has the same length as the vector and is parallel to the vector. For eample, ' is a line segment that is the same length as and is parallel to vector v. You can use these facts about parallel lines to draw translations. Parallel lines are alwas the same distance apart and never intersect. Parallel lines have the same slope. Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

3 Eample 1 Draw the image of after a translation along v. Draw a cop of v with its initial point at verte of. The cop must be the same length as v, and it must be parallel to v. Repeat this process at vertices and. Draw segments to connect the terminal points of the vectors. Label the points ', ', and '. ''' is the image of. Draw a vector from the verte that is the same length as and vector. The terminal point ' will be units up and 3 units. Draw three more vectors that are parallel from,, and with terminal points ', ', and D'. Houghton Mifflin Harcourt Publishing ompan Reflect D Draw segments connecting ', ', ', and D' to form. 3. How is drawing an image of quadrilateral D like drawing an image of? How is it different? Module Lesson 1

4 Your Turn. Draw the image of after a translation along v. Eplain Drawing Translations on a oordinate Plane vector can also be named using component form, a, b, which specifies the horizontal change a and the vertical change b from the initial point to the terminal point. The component form for _ PQ is 5, 3. You can use the component form of the vector to draw coordinates for a new image on a coordinate plane. using this vector to move a figure, ou are moving the -coordinate 5 units to the right. So, the new -coordinate would be 5 greater than the -coordinate in the preimage. Using this vector ou are also moving the -coordinate up 3 units. So, the new -coordinate would be 3 greater than the -coordinate in the preimage. Rules for Translations on a oordinate Plane Translation a units to the right, + a, Translation a units to the left, a, Translation b units up,, + b Translation b units down,, b P 5 Q 3 So, when ou move an image to the right a units and up b units, ou use the rule, + a, + b which is the same as moving the image along vector a, b. Eample alculate the vertices of the image figure. Graph the preimage and the image. Preimage coordinates: (, 1), ( 3, ), and ( 1, ). Vector:, 6 Predict which quadrant the new image will be drawn in: 1 st quadrant. Use a table to record the new coordinates. Use vector components to write the transformation rule. Preimage coordinates (, ) Image ( +, + 6) (, 1) (, 7) ( 3, ) (1, ) ( 1, ) (3, ) Then use the preimage coordinates to draw the preimage, and use the image coordinates to draw the new image. 6 Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

5 Preimage coordinates: (3, ), (, ), and (, ). Vector, 3 Prediction: The image will be in Quadrant. Preimage coordinates (, ) Image (, + ) (3, ) (, ) (, ) (, ) (, ) (, ) Your Turn Draw the preimage and image of each triangle under a translation along, Triangle with coordinates: (, ), (1, ), (, ). 6. Triangle with coordinates: P (, 1), Q (, 3), R (, 3). Houghton Mifflin Harcourt Publishing ompan Eplain 3 Specifing Translation Vectors You ma be asked to specif a translation that carries a given figure onto another figure. You can do this b drawing the translation vector and then writing it in component form. Eample 3 - Specif the component form of the vector that maps to '''. - Determine the components of v. The horizontal change from the initial point (, 1) to the terminal point (1, 3) is 1 ( ) = 5. The vertical change from the initial point (, 1) to the terminal point (1, 3) is 3 1 = Write the vector in component form. v = 5, - Module Lesson 1

6 Draw the vector v from a verte of to its image in '''. Determine the components of v. The horizontal change from the initial point ( 3, 1) to the terminal point (, ) is =. The vertical change from the initial point to the terminal point is = Write the vector in component form. v =, Reflect 7. What is the component form of a vector that translates figures horizontall? Eplain. Your Turn 8. In Eample 3, suppose ''' is the preimage and is the image after translation. What is the component form of the translation vector in this case? How is this vector related to the vector ou wrote in Eample 3? Elaborate 9. How are translations along the vectors a, b and a, b similar and how are the different? 1. translation along the vector, 7 maps point P to point Q. The coordinates of point Q are (, 1). What are the coordinates of point P? Eplain our reasoning. 11. translation along the vector a, b maps points in Quadrant I to points in Quadrant III. What can ou conclude about a and b? Justif our response. Houghton Mifflin Harcourt Publishing ompan 1. Essential Question heck-in How does translating a figure using the formal definition of a translation compare to the previous method of translating a figure? Module Lesson 1

7 Evaluate: Homework and Practice Draw the image of after a translation along v Online Homework Hints and Help Etra Practice. Line segment XY was used to draw a cop of. XY is 3.5 centimeters long. What is the length of ' + ' + '? Draw the preimage and image of each triangle under the given translation. X Y 5. Triangle: (-3, -1) ; (, ) ; (, -1) ; Vector: 3, 6. Triangle: P (1, -3) ; Q (3, -1) ; R (, -3) ; Vector: -1, 3 7. Triangle: X (, 3) ; Y ( 1, 1) ; Z ( 3, ) ; Vector:, Houghton Mifflin Harcourt Publishing ompan 8. Find the coordinates of the image under the transformation 6, -11. (, ) (, -3) (3, 1) (, -3) 9. Name the vector. Write it in component form. 1. Match each set of coordinates for a preimage with the coordinates of its image after appling the vector 3, -8. Indicate a match b writing a letter for a preimage on the line in front of the corresponding image.. (1, 1) ; (1, 1) ; (6, 5) (6, -1) ; (6, -) ; (9, -3) G H. (, ) ; (3, 8) ; (, ) ; (7, 8) (1, -6) ; (5, -6) ; (-1, -8) ; (7, -8). (3, ) ; (3, ) ; (6, 5) (, -7) ; (13, -7) ; (9, -3) D. (, ) ; (, ) ; (-, ) ; (, ) (3, -8) ; (6, ) ; (7, -8) ; (1, ) Module Lesson 1

8 11. Persevere in Problem Solving Emma and Ton are plaing a game. Each draws a triangle on a coordinate grid. For each turn, Emma chooses either the horizontal or vertical value for a vector in component form. Ton chooses the other value, alternating each turn. The each have to draw a new image of their triangle using the vector with the components the chose and using the image from the prior turn as the preimage. Whoever has drawn an image in each of the four quadrants first wins the game. Emma s initial triangle has the coordinates (-3, ), (-, ), (, ) and Ton s initial triangle has the coordinates (, ), (, ), (, 3). On the first turn the vector 6, -5 is used and on the second turn the vector -1, 8 is used. What quadrant does Emma need to translate her triangle to in order to win? What quadrant does Ton need to translate his triangle to in order to win? Specif the component form of the vector that maps each figure to its image D - D Eplain the Error ndrew is using vector v to draw a cop of. Eplain his error. 16. Eplain the Error Marcus was asked to identif the vector that maps DEF to D'E'F'. He drew a vector as shown and determined that the component form of the vector is 3, 1. Eplain his error. D - E F D E F Houghton Mifflin Harcourt Publishing ompan - Module 17 8 Lesson 1

9 17. lgebra cartographer is making a cit map. Line m represents Murph Street. The cartographer translates points on line m along the vector, to draw Nolan Street. Draw the line for Nolan Street on the coordinate plane and write its equation. What is the image of the point (, 3) in this situation? - m H.O.T. Focus on Higher Order Thinking 18. Represent Real-World Problems builder is tring to level out some ground with a front-end loader. He picks up some ecess dirt at (9, 16) and then maneuvers through the job site along the vectors -6,,, 5, 8, 1 to get to the spot to unload the dirt. Find the coordinates of the unloading point. Find a single vector from the loading point to the unloading point. 19. Look for a Pattern checker plaer s piece begins at K and, through a series of moves, lands on L. What translation vector represents the path from K to L? K L Houghton Mifflin Harcourt Publishing ompan Image redits: (t) Hero Images/Gett Images; (b) Igor Golovnov/lam. Represent Real-World Problems group of hikers walks miles east and then 1 mile north. fter taking a break, the then hike miles east to their final destination. What vector describes their hike from their starting position to their final destination? Let 1 unit represent 1 mile. 1. ommunicate Mathematical Ideas In a quilt pattern, a polgon with vertices (-, ), (-3, -1), (, ), and (-3, -3) is translated repeatedl along the vector,. What are the coordinates of the third polgon in the pattern? Eplain how ou solved the problem. Module Lesson 1

10 Lesson Performance Task contractor is designing a pattern for tiles in an entrwa, using a sun design called Image for the center of the space. The contractor wants to duplicate this design three times, labeled Image, Image, and Image D, above Image so that the do not overlap. Identif the three vectors, labeled m, n, and p that could be used to draw the design, and write them in component form. Draw the images on grid paper using the vectors ou wrote. Image Houghton Mifflin Harcourt Publishing ompan Module 17 8 Lesson 1