3.1 Sequences of Transformations Per Date Pre-Assessment Which of the following could represent a translation using the rule T (, ) = (, + 4), followed b a reflection over the given line? (The pre-image is the shaded circle.) Note: there could be more than one correct answer. a) b) c) d) Which of the following could represent the figure, after a vertical translation b a non-zero value followed b a reflection over a vertical line? (The pre-image is the shaded triangle.) a) b) c) d) 1. Toda we will be learning more about rigid motion transformations, which are sequences of basic transformations. The two eercises above are eamples of sequences of rigid motion transformations. Eplain what ou think sequences of rigid motion transformations are. Page 1
3.1 Sequence of Transformations Per Date Grandma decides she wants to rearrange her room and wants to tr it out on paper before asking her grandchildren to help her move everthing. Identif the objects in the room that could be moved to their new position as a result of a single transformation. Original Room Laout Rearranged Room Laout In this lesson we will eplore sequences of transformations, also called composite transformations. Later we will return to Grandma s room and identif the transformations required to complete the rearrangement of her room. Page 2
3.1 Sequence of Transformations Per Date One of the fundamental questions in is How can we tell if Figures A and B have the same shape and size, without measuring them? Intuitivel, this could be answered es if it were possible to Cut out Figure A, move it around and possibl flip it over, and place it onto Figure B so that it matches perfectl. In the last lesson we learned how to translate, reflect, and rotate figures in the plane, which clearl resulted in images that retained the same size and shape as their pre-image. But, not all images with the same size and shape result from using onl a single simple transformation; sometimes moving Figure A onto Figure B requires using a sequence of simple transformations. For eample, we ma need to translate Figure A, then rotate it, and if the orientation is still off reflect it as well. Such a sequence of translations, reflections, and rotations is referred to as a rigid motion transformation. Wh do ou think the call these Rigid Motion? Rigid Motion Transformation (#VOC) a (or sequence) of one or more transformations (,,. Rigid motion transformations preserve distances (i.e. size) and angle measures (i.e. shape). Note: a single translation, rotation, or reflection is a simple eample of a rigid motion transformation. 1. For each of the following pairs, the left pre-image was transformed into the right image. Which CANNOT be achieved with a single translation, rotation, or reflection? Argue wh it is not possible. You do not need to identif a transformation. Page 3
3.1 Sequence of Transformations Per Date 2. Translate ABC to the right 9 units and down 7 units and label the image A B C. Translate the image triangle A B C ou just drew to the left 2 units and up 3 units, label this image A B C. Describe the single translation that would carr ABC onto A B C. The above eample can be summarized with one single translation. Therefore, this is not considered a sequence of transformations. 3. Translate DEF to the right 4 units and down 2 units. Label the image D E F. Reflect D E F over the horizontal line and label the image D E F. Notice that this transformation CANNOT be achieved with a single transformation. Therefore this is considered a sequence of transformations, and D E F is congruent to DEF. The angle measurements and side lengths of the image D E F are equal to those of the pre-image, DEF. Page 4
3.1 Sequence of Transformations Per Date Practice: 1. Translate ΔRST using the rule T (, ) = ( 4, 3), then reflect over the -ais. Draw and label the image ΔR S T. 2. Reflect ΔRST over the -ais, then reflect over the line = 3. Draw and label this image ΔR S T. 3. Point P is in quadrant III and is then reflected over the -ais followed b a reflection over the -ais. In which quadrant is the new P? a) Quadrant I b) Quadrant II c) Quadrant III d) Quadrant IV 4. Which of the following diagrams could represent a reflection over a vertical line followed b a non-zero vertical translation? (The pre-image is the shaded star.) A B C D Page 5
3.1 Sequence of Transformations Per Date Assume all of the points below are graphed on a Cartesian Coordinate Plane. Give the new coordinates after performing the rigid motion transformation. 5. Point A (2, 6) is translated horizontall 4 units and verticall 6 units, then reflected over the -ais. Give the coordinates for A. 6. Point B ( 3, 7) is reflected over the -ais, then translated verticall 8 units. Give the coordinates for B. 7. Point C ( 22, 17) is translated using the rule: T (, ) = ( 13, + 4), then reflected over the -ais. Give the coordinates for C. 8. Point X (1, 2) was transformed to point X ( 1, 2). What tpe of transformation could NOT have occurred? a) a 180 rotation about the origin b) a reflection over the -ais followed b a reflection over the -ais c) a vertical translation of 4 followed b a reflection over the -ais d) a horizontal translation 2 followed b a reflection over the -ais e) none of these all transformations would give the same result Page 6
3.2 - Composite Transformations Per Date Indicate for each of the three following pairs of transformations, whether the two composite transformations would sometimes, alwas, or never give the same result when applied to a figure. Justif our answer b eplaining in words or drawing diagrams. 1. Would the two composite transformations give the same result alwas, sometimes, or never? Defend our answer. A reflection over the -ais followed b a vertical translation of 6. A vertical translation of 6 followed b a reflection over the -ais. 2. Would the two composite transformations give the same result alwas, sometimes, or never? Defend our answer. A translation, T (, ) = (, + 2) followed b a reflection over the -ais. A reflection over the -ais followed b another reflection over the -ais. 3. Would the two composite transformations give the same result alwas, sometimes, or never? Defend our answer. A reflection over the -ais followed b a reflection over the -ais. A reflection over the -ais followed b a reflection over the -ais. Page 7
3.2 Composite Transformations Per Date Scavenger Hunt: How it works 1. At our first poster, ignore the composite transformation and letter (these go with another description). 2. Write down the description/rule from the bottom of the poster in the first bo below. 3. Draw a sketch below of the composite transformation described b in the poster. 4. Look around the room for a diagram that matches our sketch and go to that poster. 5. Write the letter from the top right corner of the new poster. 6. Find our net description/rule at the bottom of the poster. 7. Repeat this process until all 10 boes are complete. 8. Write out the letters in order to reveal a new description in how to transform a figure for our last eercise! 1) Description/Rule: 2) Description/Rule: 3) Description/Rule: 4) Description/Rule: Page 8
3.2 Composite Transformations Per Date 5) Description/Rule: 6) Description/Rule: 7) Description/Rule: 8) Description/Rule: 9) Description/Rule: 10) Description/Rule: Page 9
3.3 Congruence Per Date We now return to our earlier question: How can we tell if it possible to cut out Figure A, move it around and possibl flip it over, so that it matches Figure B perfectl, retaining the size and shape of Figure A? 1. Working on our own, use patt paper to trace Figure A on the left below (or phsicall cut it out if ou don t have patt paper). Then, move it around so that ou can place it in a perfectl matching manner on top of Figure B to the right. Keep track of the tpes of translations, rotations, or reflections that ou use. Tr to onl use at most one of each. Label the vertices of the image appropriatel as A, B, and C. Describe in words the simple transformations ou used. Tr to be as eplicit as possible. Now compare our work with a partner. Did ou use the same rigid motion transformation? If not, could ou have used a different rigid motion transformation? How do ou know the side lengths and angles of the image are equal to those of the pre-image? Page 10
3.3 Congruence Per Date 2. Use a ruler and protractor to confirm that the sizes and shapes of the two triangles shown below are identical. Since these two triangles have the same size and shape, there should be a wa to cut out the one on the left and move it around to match the one on the right. Therefore, there must be a rigid motion transformation that does just that. Describe a rigid motion transformation with pre-image ABC and image A B C. Use patt paper or a cut-out first, if ou like. Work with a partner to tr to determine a sequence of steps that will alwas work for identifing a rigid motion transformation that will move a pre-image triangle to a matching image, as long as the image has the same size and shape. Your steps should work regardless of the location of the image triangle. Page 11
3.3 Congruence Per Date 3. For this problem ou will work both on our own and with a partner. Complete steps a c and e on our own. a. Trace or cut out a cop of the triangle below. b. Write down a sequence of translations, rotations, and/or reflections, being sure to keep our rigid motion transformation secret from our partner. Make sure the resulting image fits on the diagram below. Adjust accordingl. You ma use whatever units ou like for the grid. c. Place the triangle cop on the grid below corresponding to the image of our transformation, and trace its location. d. Echange our diagram with our partner. Be sure to indicate the units ou used. e. Identif a rigid motion transformation that has the image our partner drew. f. Share our results with our partner, and check to see if our partner s rigid motion transformation ields the image ou traced. g. Did our partner use the same transformation ou wrote down in secret? Don t worr if the are not the same, there are multiple transformations that ield the same image. Page 12
3.3 Congruence Per Date In this lesson we have seen that the image of a rigid motion transformation has the same size (e.g. triangle side lengths are equal) and shape (e.g. angles measures are equal) as its pre-image. We have also seen that if two figures have the same size and shape, then there eists a rigid motion transformation from one onto the other. Thus, we now have an answer to our earlier question: How can we tell if Figures A and B have the same size and shape without measuring (or cutting out and moving)? Answer: Figure A has the same size and shape as Figure B onl if there eists a rigid motion transformation with pre-image Figure A and image Figure B (or vice-versa since translations, rotations, and reflections are all reversible). In such a case we sa that Figure A is congruent to Figure B, and we use the notation A B. Figure A is said to be congruent (#VOC) to Figure B if there eists a rigid motion transformation with pre-image Figure A and image Figure B. This is the formalization of being able to cut out Figure A and place it perfectl onto Figure B. Finding a Rigid Motion Transformation for Triangles Summar The following steps make it easier to find the rigid motion transformation from one triangle to another, assuming the are congruent. (Note: this method can be adjusted for other shapes as well.) i. Choose a verte, sa A, on the pre-image triangle. ii. iii. iv. Identif the corresponding verte, sa A, on the image triangle. (Note: since the vertices are not labeled in most cases, it is up to ou to visuall determine corresponding vertices b their equal angle measures.) Translate the pre-image so that A moves to A. Rotate around A as needed. v. If the orientation is still off, reflect about the line = A or = A depending on the orientation. Note: steps iii - v can be switched per our preferences, and ou ma not need to use all three tpes of transformations. Let s tr out our new method! Page 13
3.3 Congruence Per Date Show that each pair of triangles below are congruent b describing a rigid motion transformation from the left triangle onto the right triangle. Be sure to label corresponding vertices A, B and C. Tr to use no more than one of each tpe of simple transformation. 4. 5. Page 14
3.3 Grandma s Furniture Pair & Share Per Date Now let s revisit Grandma s rearrangement of her room. Identif the sequence of transformations or single transformation that could be used to move each object in the room. Original Room Laout Rearranged Room Laout Lamp Chair Couch Table Desk Fan Page 15