ࡱ> ln ghijkVlpr~ ~ c bjbj .W\W\0 ^ ^ _______zc _MTvo zzzz;H;H;H$W_;H5; ;H;H;Hzzyyy;H4z_zy;Hyyh|\K6|z kOJ|0M|,IZ|_|;H;Hy;H;H;H;H;Hgj";H;H;HM;H;H;H;H;H;H;H;H;H;H;H;H;H^ > : Geometry Assignments: Transformations DayTopicsHomeworkHW GradeQuiz Grade1Line reflections HW TG - 1 2RotationsHW TG - 2 3Translations HW TG - 3 4Practice **QUIZ**HW TG - 4 5DilationsHW TG - 5 6CompositionsHW TG - 6 7Practice **QUIZ**HW TG - 78Rigid motions HW TG - 89Rigid Motions and CongruenceHW TG - 910Review **QUIZ**HW TG - Review11**TEST** Geometry: Transformations Answers HW - 1 1b. Construct the perpendicular bisectors of the segments joining each vertex to its image (the ( bis. of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and EMBED Equation.DSMT4 ). If they are all the same line, its a reflection. 3a. (2, 7) b. (7, 2) c. (2, 7) 4. (3, 1) 5. (4, 5) 6. x = 2 7a. B b. (0, 3) and (0, 4) c. (2, 2) and (4, 4) 8a. x = 5 b. y = x 2 c. y = 2.5 9. a and b 10c. (4, 1) 11. x = 1, y = 2, y = x + 1, y = x+ 3 HW - 2 2. (1, 3) 3a. (4, 5) c. Nothing 4. (2, 5) 5a. P'(x, y) b. P"(x, y) c. Ro(x, y) 6. a and e are rotations 7a. Clockwise; same. b. a and e are same; rest are opposite c. A rotation preserves orientation (keeps it the same). A reflection changes orientation to the opposite. 8c. y = 0.5x 2 d. y = 2x 4 10a. 45( CCW b. 4; rotations preserve distance c. 90(; rotations preserve angle measure 10a. (3, 4) b. (4, 3) c. (2, 0) d. (3, 2) e. (7, 5) 11a. 18( b. 15 (pentadecagon, another word you dont need to memorize) HW - 3 1. (5, 1) 2. (3, 4) 3. T2, 3(x, y) or (x + 2, y 3) 4a. (0, 2) b. (1, 1) 5. A'(1, 0), B'(3, 3), C'(6, 1) 6a. 5 b. 3/4 7d. T6, 2 8d. T6, 0 9. (x + a, y + b) b. b/a c.  EMBED Equation.DSMT4  9a. Reflection over  EMBED Equation.DSMT4  b. 180( rotation around M c. Reflection over ( bisector of  EMBED Equation.DSMT4  d. 90( rotation around M e. Translation along vector from A to C f. Reflection over ( bisector of  EMBED Equation.DSMT4  HW 4 2a. a = 2, b = 1 3a. 180( around midpoint of  EMBED Equation.DSMT4  b. Reflection over ( bisector of  EMBED Equation.DSMT4  4a. n b. 360/n 5. c and d are not preserved 6. c is not preserved 7a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c. Distance is preserved in rotations 8a. (APQ ( (BPQ b. Angle measure is preserved in reflections c.  EMBED Equation.DSMT4  d. Distance is preserved in reflections 10. a. 1) Translate along the vector  EMBED Equation.DSMT4  so A goes to O and C goes to D  2) Reflect over  EMBED Equation.DSMT4  so T goes to G b. 1) Translate along the vector  EMBED Equation.DSMT4  so A goes to O 2) Rotate 90( CCW around O so T goes to G and C goes to D HW - 5 1. (8, 4) 2. (6, 15) 3. (2, 6) 4. 4 5. 2/3 6a. S b. P c. O d.  EMBED Equation.DSMT4  e. 1/2 f. 12 7. (3) 8a. 4 b. 12 c. 6 9c.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  d. 3 e. 3, 27 f. 9 10a. P = 90, A = 486 b. P = 15, A = 13.5 11e. same, parallel HW - 6 1. (9, 12) 2. (5, 4) 3. (2, (4) 4. ((4, 1) 5. R90( 6. D6 7a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c. No 8a.  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  (there are others) b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  9. 1) Translate along vector  EMBED Equation.DSMT4  2) Rotate CW until  EMBED Equation.DSMT4  coincides with  EMBED Equation.DSMT4  3) Reflect over  EMBED Equation.DSMT4  10d. Yes. When the translation is parallel to the line of reflection. HW - 7 1. (3, 4) 2. (1, 3) 3. (15, 6) 4. (6, 3) 5. (4, 2) 6. (4, 5) 7. (4, 2) 8. (3, 2) 9. (8, 2) 10. (2, 1.5) 11. (2, 6) 12. (4, 5) 13. k = 4/3 14a. D3 b. T4,8 c. (4, 8) d.  EMBED Equation.DSMT4  15. (1, 2) 16. y = 4 17a. (0, 2) b. (1, 1) 18a. 45( b. 15 19.  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  (There are other ways.) 20. 1) Translate along vector  EMBED Equation.DSMT4  2) Rotate CW until  EMBED Equation.DSMT4  aligns with  EMBED Equation.DSMT4  3) Reflect over  EMBED Equation.DSMT4  HW - 8 1a. A transformation that preserves distance: translations, (line) reflections and rotations. b. Yes, yes, yes, yes, no, no. 2. (5) 3. (1) 4a. (O) b. (D) c. (D) d. (N) e. (O) f. (N) g. (D) h. (D) i. (D) j. (O) 5. a, c, f and h are rigid motions. b, g. Distance and angle measure not preserved d. Changes shape e. Lines not preserved 6a. 90( rotation CCW followed by a translation. b. 90( rotation CW followed by vertical line reflection and (if necessary) a translation. c. 180( rotation (either way) followed (if necessary) by a translation. d. 45( rotation CCW followed by a horizontal line reflection and (if necessary) a translation. Note: all of the above have other possible answers. 6a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  Again, other answers are possible. HW - 9 1. They must have measure (length). 2. They must have the same measure (degrees). 3a. (DCE ( (BAF, (CED ( (AFB, (EDC ( (FBA b.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  4. No (consider two rectangles of different sizes) 5. No (consider a square and a non-square rhombus) 6a. (CBM b. Reflection over line BM. 7a. (RTS b. Reflection over horizontal line followed by translation 8a. LOVE b. 180( rotation followed (if necessary) by translation 9a. RQPS b. CCW rotation til  EMBED Equation.DSMT4 , then vertical line reflection, then (if necessary) translation. 10. 60 11. 120( Review Answers 1. (2, 3) 2. (6, 8) 3. x = 3 4. (2, 4) 5. (4) 6. (3) 7. (1) 8. (1) 9. (1 ) 10. (2) 11. (2) 12. (4) 13. (4) 14. (1) 15. (1) 16. (3) 17. (4) 18.  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  19. (6, 3) 20a. 3/4 b. 5 c. 3/4 d. 30 21a. 5/2 b. 10 c. 12 22. 6 23a. x = 1, y = 2, y = x + 1, y = x + 3 b. 90(, 180(, 270( or 360( around (1, 2) 24a. 31.5 b. 47.25 25a. Reflect over y = 1 then translate right 4. b. Rotate 90( CW about the origin, reflect over the x-axis, then translate down 3 OR Rotate 90( CW about the origin, then reflect over the line y = 1.5 OR Reflect over the line y = x then translate down 3 c. Dilate by factor of 2 centered at the origin, rotate CCW 90( about the origin, then translate left 1, up 2 OR Rotate CCW 90( about the origin, translate left 1, up 2, then dilate by factor of 2 centered at (1. 2) 26. 1) Translate so B maps A. 2) Rotate CW around A until I maps to T. 3) Reflect over  EMBED Equation.DSMT4  so X maps to R and E maps to H EMBED Equation.DSMT4  EMBED Equation.DSMT4  Geometry HW: Transformations - 1 Name  1. Consider the diagram at right of YAK and Y'A'K'. a. Suppose Y'A'K' is the image of YAK after a line reflection. Use a compass and straight edge to construct the line of reflection. HINT: Consider property 2 of a reflection. b. Suppose we are not sure if Y'A'K' is the image of YAK after a line reflection. Describe briefly but precisely how we could find out if it is. 2. Construct  EMBED Equation.DSMT4 , the reflection of  EMBED Equation.DSMT4  over line k.      3. Find the coordinates of the image of the point (2, 7) under each of the following: a. ry-axis. b. ry = x. c. rx-axis. 4. If the point (3, 1) is reflected in the x-axis and then that image is reflected in the y-axis, what are the coordinates of the final image? 5. What are the coordinates of the image of the point (4, 1) after a reflection in the line y = 3? 6. The image of the point A(3, 1) after a reflection in line k is (7, 1). Find the equation of line k.  7. Triangle ABC is shown in the graph at right. a. Which point on the triangle will be invariant under a reflection in the x-axis? b. Give the coordinates of the points on the triangle that will be invariant (unchanged) under a reflection in the y-axis. (Invariant points are often called fixed points.) c. Give the coordinates of the points on the triangle that will be invariant under a reflection in the line y = x. 8. For each diagram below, (C'A'T' is the image of (CAT after a line reflection. Write the equation of the line of reflection.  a. b. c. 9. In which of the following is (A'B'C' not the image of (ABC after a reflection?  a. b. c. d. 10. a. Graph the line l having equation  EMBED Equation.DSMT4  and the point A(8, 9). b. A' is the image of A after a reflection over line l. What is the slope of  EMBED Equation.DSMT4 ? Why? c. Find the coordinates of A'. Read: Line Symmetry  A figure has line symmetry if it is its own image after a line reflection. Alternate wording (popular on the Regents): a figure maps onto itself after a reflection over a line of symmetry. In the rhombus shown at right, lines ( and k are lines of symmetry; line n is not. 11. A square is graphed at right. Write equations for all the lines of symmetry for this square. Geometry HW: Transformations - 2 Name 1. Segment AB is to be rotated counterclockwise about center P by the angle (. a. Use a compass to locate A', the image of A.  b. Use a compass and straight edge to locate B', the image of B. (There is more than one way to do this. The easiest is to remember that rotations preserve distance. So A'B' = AB and P'B' = PB.) 2. What is the image of (3, 1) under a rotation of 90 about the origin? 3. a. What is the image of (4, 5) under a rotation of 180 about the origin? b. What is the difference between this and a rotation of 180( (i.e. 180( CW) about the origin? 4. Using the rule (x, y) ( (y, x), find the image of A(5, 2). 5. a. Find the coordinates P', the image of P(x, y) after a reflection in the x-axis. b. Find the coordinates P", the image of P' after a reflection in the y-axis. c. A reflection in the x-axis followed by a reflection in the y-axis is the same as what single transformation?  6. In each of the following, tell whether the shaded triangle is the image of the unshaded triangle after a reflection (r) or a rotation (R). a. b. c. d. e. f. 7. a. Review problem #6a. Starting at A in (ABC and going in alphabetical order around the triangle, do the points go clockwise or counterclockwise? This is called the orientation of the figure. Is the orientation of the image, (A'B'C, the same or the opposite? b. For each of 6b 6f, determine if the orientation of the image is the same or the opposite of the original. (Note: In each figure, the original is clockwise.) c. How can we use orientation to tell if a figure has been reflected or rotated?  8. a. Graph the line (: y = 0.5x + 2. b. Graph line k, the image of line ( under a reflection in the origin. One way you can do this: Pick three points on ( and find their images. c. Write an equation for line k. d. Graph line n, the image of line ( after a 90( rotation about the origin. e. Write an equation for line n.  9. a. Graph DRAT having coordinates R(0, 2), A(2, 5) and T(7, 2). b. Graph DR'A'T', the image of DRAT after a 90( rotation about the origin. c. Graph DR"A"T", the image of DRAT after a reflection in the line y = x. 10. In the diagram at right, (R'A'T' is the image of (RAT after a rotation around point P. a. What is the angle and direction of rotation? (You do not need a protractor, just your brain.) b. What is the length of  EMBED Equation.DSMT4  ? How do we know? c. What is the measure of (R'T'A' ? How do we know? Read: Rotational Symmetry and Point Symmetry A figure has rotational symmetry if it is the image of itself (maps onto itself) after a rotation of 0( < ( < 360(.  Ex: Ex:  11. a. What is the minimum angle of rotation that will take a regular icosagon onto itself? (An icosagon is a 20-sided polygon; that is not a vocab word you need to remember.) b. A regular polygon has a minimum angle of 24( to carry it onto itself. How many sides does it have? Geometry HW: Transformations - 3 Name 1. Use a compass and straight edge to translate triangle (ABC the distance and direction of the directed line segment (aka vector) PQ.  2. Find the image of the point (3, 5) under the translation (x, y) ( (x + 2, y 4).  3. Under a certain translation, the image of A is R. What two things must be true about segments  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 for us to be sure that under that same translation the image of B is S? 4. Under a given translation, the image of (4, 2) is (6, 1). a. Find the image of (2, 5) under the same translation. b. Find the preimage of (3, 4) under the same translation.  5. In the diagram at right, (ABC has vertices A(2, 2), B(0, 5) and C(3, 1). Vector  EMBED Equation.DSMT4  has initial point P(1, 1) and terminal point Q(2, 3). State the coordinates of the vertices of (A'B'C', the image of (ABC after the transformation  EMBED Equation.DSMT4 . 6. Under the translation P(x, y) ( P'(x + 4, y + 3), a. What is the distance between any point P and its image P' ? b. What is the slope of the line PP' ?  7. a. Graph y = x2 + 3 for 2 ( x ( 2. b. Draw the image of the graph in part a after a translation T2, 5. c. Draw the image of the graph in part b after a translation T4, 3. d. Name the single transformation that is equivalent to T2, 5 followed by T4, 3.  8. In each of the following, the dashed triangle is the image of the solid triangle after a single transformation. Name the transformation. Be specific (e.g., Reflection over line _______; not just reflection).  a. b. c.   d. e. f. Geometry HW: Transformations - 4 Name  1. a. Graph the circle C having equation  EMBED Equation.DSMT4 . b. Graph circle C', the image of circle C after a reflection in the x-axis. c. Graph circle C'', the image of circle C after a reflection in the line y = x. d. Graph circle C''', the image of circle C after a 90 rotation about the origin. e. Graph circle C*, the image of circle C after a translation along the vector  EMBED Equation.DSMT4 . 2. Given points A(2, 4) and B(6, 2). What are the values of a and b such that the translation Ta, b maps A onto the midpoint of  EMBED Equation.DSMT4 ? 3. Let  EMBED Equation.DSMT4  be a line segment. a. What rotation would map  EMBED Equation.DSMT4  onto itself? (Be specific: what point and what angle?) b. What reflection (other than over itself) would map  EMBED Equation.DSMT4  onto itself? (Again, be specific: what line?) 4. A regular polygon is a polygon where all sides are congruent and all angles are congruent (equilateral triangle, square, etc.). a. How many lines of symmetry does a regular polygon with n sides have? b. What is the degree measure of the smallest rotation that will map a regular polygon with n sides onto itself? READ: Orientation refers to the direction in which points go around a figure. In two-dimensional geometry, the only orientations are clockwise and counter-clockwise.  (CAT is oriented clockwise. (DOG is oriented counter-clockwise.  5. Trapezoid ABCD is reflected over line  EMBED Equation.DSMT4 . Tell if each of the following is preserved (stays the same) or not. a. The length of  EMBED Equation.DSMT4 . b. The measure of (D c. The slope of  EMBED Equation.DSMT4  d. The orientation of ABCD e. The area of ABCD f. The parallelism of  EMBED Equation.DSMT4  and EMBED Equation.DSMT4 . 6. Parallelogram ABCD is rotated 90 around point A. Tell if each of the following is preserved or not.  a. The length of diagonal  EMBED Equation.DSMT4 . b. The measure of (B c. The slopes of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  d. The parallelism of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  e. The orientation of ABCD f. The area of DABC  7. In the diagram at right,  EMBED Equation.DSMT4  is rotated clockwise around point P by an amount equal in measure to (APB. a. What is the image of  EMBED Equation.DSMT4 ? b. What must be true for the image of A to be B? d. If  EMBED Equation.DSMT4  what property of rotations guarantees that the image of A will be B?  8. In the diagram at right,  EMBED Equation.DSMT4  is reflected over  EMBED Equation.DSMT4 . a. What must be true for the image of  EMBED Equation.DSMT4  to be  EMBED Equation.DSMT4 ? b. If (APQ ( (BPQ what property of reflections guarantees that the image of  EMBED Equation.DSMT4  will be  EMBED Equation.DSMT4 ? c. In addition to the above, what must be true for the image of A to be B? d. If  EMBED Equation.DSMT4  what property of reflections guarantees that the image of A will be B?  9. In the diagram, points A and B are on line l, which is neither horizontal nor vertical.  EMBED Equation.DSMT4  is horizontal and  EMBED Equation.DSMT4  is vertical, making (ABC a right triangle. Line l' and triangle AB'C' are the images of line l and (ABC after a 90( rotation about A. a. What is the measure of (BAB' ? b. What is the relationship between lines l and l' ? c. Give the lengths of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Justify your answers. d. What is the slope of line l? What is the slope of line l' ? e. How does the slope of line l' compare to the slope of line l (two words)? 10. In each of the following, describe in words a sequence of two transformations after which (DOG is the image of (CAT. Note: There are several possible correct answer to each.  a. b. Geometry HW: Transformations - 5 Name 0. Using a compass and straight edge, construct the image of (ABC after a dilation of scale factor 3 centered at point P.   1. In the diagram at right, O is the center of dilation and Dk(DOQR) = DOPS. a. What is the image of R under the dilation? b. Dk(Q) = c. Dk(O) = d. Dk EMBED Equation.DSMT4  = e. If P is the midpoint of  EMBED Equation.DSMT4 , what is the constant of dilation k? f. Using the value of k from part e, if SP = 6, find RQ. 2. a. In the diagram,  EMBED Equation.DSMT4  is the image of  EMBED Equation.DSMT4  after a dilation of scale factor k with center P. Which ratio is equal to the scale factor k of the dilation? (Multiple choice.) (1)  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4  (3)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  b. In the same diagram, suppose  EMBED Equation.DSMT4  is the image of  EMBED Equation.DSMT4  after a dilation of scale factor k with center P. Which ratio is equal to the scale factor k of the dilation? (Multiple choice.) (1)  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4  (3)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  3. In the diagram (which is NOT to scale), (PQR is the image of (ABC after a dilation of scale factor k with center P.  a. What is the scale factor of the dilation? (Think hard about this.) b. What is the length of  EMBED Equation.DSMT4 ? c. What is the length of  EMBED Equation.DSMT4 ?  4. In the diagram, rectangle BCDE is the image of rectangle ABFG after a dilation centered at O. OA = 4, AB = 6, and AG = 8. a. What was the scale factor of the dilation? b. What re the dimensions, base and height, of rectangle BCDE?  5. a. Graph DABC with vertices A(1, 3), B(4, 1), and C(1, 1). b. Graph DA'B'C', the image of DABC after a dilation of scale factor 3 centered at the origin. c. Find the lengths of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  in simplest radical form. d. How many times longer is  EMBED Equation.DSMT4  than  EMBED Equation.DSMT4 ? e. Find the areas of DABC and DA'B'C'. f. How many times larger is the area of DA'B'C' than the area of DABC? 6. A certain hexagon has a perimeter of 30 and an area of 54. a. Find the perimeter and the area of the hexagon after a dilation of 3. b. Find the perimeter and the area of the hexagon after a dilation of 1/2. 7. a. Graph the line l, EMBED Equation.DSMT4 .  b. Graph the image of l after a dilation of 3 in the origin. What is the equation of the image? c. Graph the line k,  EMBED Equation.DSMT4  d. Graph the image of k after a dilation of 3 in the origin. What is the equation of the image? 8. If the line y = mx + b is dilated by a scale factor of k centered at the origin, in the image (1) Both the slope and the y-intercept will be multiplied by k. (2) The slope will be multiplied by k but the y-intercept will be unchanged. (3) The y-intercept will be multiplied by k but the slope will be unchanged. (4) Neither the slope nor the y-intercept will be changed.  9. Line ( is shown at right. a. What will be the image of line ( after a dilation of scale factor 2 centered at point P? b. Graph the image of line ( after a dilation of scale factor 2 centered at point Q. (Pick two convenient points on the line and find the image of each. Two points determine a line.) c. How does the image of line ( in part b compare to the preimage? 10. Line segment  EMBED Equation.DSMT4  , whose endpoints are (6, 4) and (3, 8), is the image of  EMBED Equation.DSMT4  after a dilation of  EMBED Equation.DSMT4  centered at the origin. What is the length of  EMBED Equation.DSMT4  ? (1)  EMBED Equation.DSMT4  (2) 5 (2) 15 (4) 45 Geometry HW: Transformations 6 Name Evaluate each of the compositions in #1 4: 1. ry-axis o D3(3, 4) 2. RO o T2, 3 (3, 1) 3. rx-axis o ry = x (4, 2) 4. R90 o ry = 3 (1, 2) Use your knowledge of transformations and common sense to answer #5 - 7 5. The composition R30 o R60 is equivalent to what single transformation? 6. The composition D2 o D3 is equivalent to what single transformation?  7. The graph shows (ABC and (DEF. a. Let (A'B'C' be the image of (ABC after a rotation about point C. State the location of B' if the location of point A ' is (5, 3). b. Name a second transformation that will take (A'B'C' onto (DEF.  8. Triangle QCK undergoes thecomposition  EMBED Equation.DSMT4 . a. Aaron does the composition correctly. Graph (Q'C'K', Aarons image of (QCK. b. Name the single transformation for which the image of (QCK would be (Q'C'K'. c. Zyta does the composition backwards. Graph (Q"C"K", Zytas image of (QCK. d. Name the single transformation for which the image of (QCK would be (Q"C"K". e. Name the single transformation for which the image of (Q"C"K" would be (Q'C'K'.  9. (ABC is graphed at right along with three transformations of it. For each triangle, a, b and c, write a composition of transformations for which the given triangle would be the image of (ABC. Note: There is more than one right answer for each of these. Try to keep it simple. a. b. c. 10. Given that TRAP and T'R'A'P' are congruent in the diagram, describe precisely a sequence of transformations that would map quadrilateral TRAP onto T'R'A'P'.  11. a. Graph the points (0, 2), (1, 1), (3, 2) and (2, 3). Connect them as shown at right: (To the unimaginative, that may look like a lopsided arrow but its actually a dinosaur footprint.) b. Draw the image of the figure from part a after the composition  EMBED Equation.DSMT4 . Note: We are only interested in the final image. If you need an intermediate step, consider making it a different color or dashed/dotted or erasing it when youre done. c. Repeat part b three more times, each time starting with the previous image. (It should get easier as you go along.) d. Is the composition  EMBED Equation.DSMT4  commutative? In general, when will the composition of a line reflection and a translation be commutative? Note: This composition is an example of a glide reflection. This is a term that does not seem to appear in the Common Core so you probably dont need to worry about it.  Geometry HW: Transformations - 7 Name 1. Use compass and straight-edge to construct the image of the triangle shown after a translation along vector PQ.  2. Use compass and straight-edge to construct the image of the triangle shown after a reflection over line l.  3. Use compass and straight-edge to construct the image of the triangle shown after a counterclockwise rotation of m(A around point P.  5. The image of the point A(2, 7) after a reflection in line k is (8, 1). Find the equation of line k.  6. In the diagram, (RST is the image of (ABC after a dilation centered at P. Which ratio does not represent the scale factor of the dilation? (1)  EMBED Equation.DSMT4  (3)  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  7. a. What is the minimum rotation needed to carry a regular octagon onto itself? b. A regular polygon has a minimum rotation of 24( to carry the polygon onto itself. How many sides does the polygon have?  8. In the diagram at right, (JKL is the image of (ABC after a dilation centered at the origin. The coordinates of the vertices are A(3, 0), B(4, 5), C(6, 0), J(5, 0),  EMBED Equation.DSMT4  and L(10, 0). Which of the following is the ratio of the lengths of  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 ? (Note: there is an easy way to do this and a hard way. Try to find the easy way.) (1)  EMBED Equation.DSMT4  (3)  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  9a. The line 2x + 3y = 12 is dilated by a scale factor of 3 centered at the origin. What is the equation of the image of the line? b. The same line is dilated by a scale factor of from the point (9, 2). What is the equation of the image? 10. Give a precise composition of transformations what will take the letter R at right onto its image. 11. Describe precisely a sequence of transformations that will take (ABC below onto its image (XYZ.  Geometry HW: Transformations - 8 Name 1. On the grids below, triangle P is the pre-image for all transformations. Match the correct image with each of the transformations below. Images may be used more than once or not at all. a. ry-axis b. ry = 1 c. ry = x d. R180( e. T4, 0 f. R90( g. rx = 1 h. T0, 10 i. R270( j. rx-axis k. R90(, (5, 5) l. R180(, (1, 5) m. R90(, (3, 3) n. rx-axis ( ry-axis o. ry = 5 ( rx-axis p. R90(,0 (1, 5) ( T4, q. R90(, (1, 5) ( T4, 0 r. ry-axis ( ry = x  2. a. What is a rigid motion? Name the three basic ones. b. In a rigid motion, 1) Is collinearity always preserved? 2) Is angle measure always preserved? 3) Is parallelism always preserved? 4) Are midpoints always preserved? 5) Is orientation always preserved? 6) Is slope always preserved? 3. Which of the following transformations is not a rigid motion? (1) Line reflection (2) Rotation (3) Translation (4) Dilation 4. Which of the following transformations does not preserve orientation? (1) Line reflection (2) Rotation (3) Translation (4) Dilation 5. Tell whether each figure represents a rigid motion.  a. b. c. d.   e. f. g. h. 6. In each diagram below, identify a specific rigid motion that would take (ABC onto its image as shown. As usual, there can be more than one right answer.  a. b. c.  d. e. f. Geometry HW: Transformations - 9 Name 1. What is necessary for two line segments to be congruent? 2. What is necessary for two angles to be congruent?  3. In the diagram at right, DCDE @ DABF. a. Name three pairs of congruent angles. b. Name three pairs of congruent sides. 4. If all four pairs of corresponding angles of two quadrilaterals are congruent, must the quadrilaterals be congruent? Draw a diagram to justify your answer. 5. If all four pairs of corresponding sides of two quadrilaterals are congruent, must the quadrilaterals be congruent? Draw a diagram to justify your answer.  6. The two triangles shown at right are congruent. a. Complete the congruence statement: (ABM ( b. Describe a rigid motion that would take the first figure onto the second.  7. The two triangles shown at right are congruent. a. Complete the congruence statement: (JKL ( b. Describe a rigid motion that would take the first figure onto the second.  8. The two quadrilaterals shown at right are congruent. a. Complete the congruence statement: MATH ( b. Describe a rigid motion that would take the first figure onto the second.  9. The two quadrilaterals shown at right are congruent. a. Complete the congruence statement: ABCD ( b. Describe a rigid motion that would take the first figure onto the second. 10. In the diagram, DABC ( DDEF. If AB = 2x + 6, BC = 3x + 2, CA = 5x  8 and EF = 4(x  1), find the numerical value of the perimeter of (ABC.  11. In the diagram (not drawn to scale), DBIG ( DCAT. Find the numerical measure of (T. Geometry HW: Transformations - Review Name 1. Using a compass and straight edge, construct the image of  EMBED Equation.DSMT4  after a reflection over line (.   2. Using a compass and straight edge, construct the image of  EMBED Equation.DSMT4  after a translation that maps A to A'.     3. Using a compass and straight edge, construct the image of  EMBED Equation.DSMT4  after a rotation about point P that maps  EMBED Equation.DSMT4 onto  EMBED Equation.DSMT4   4. Pont A' is the image of point A after a line reflection. Using a compass and straight edge, construct the line of reflection.  5. Segment A'B' is the image of  EMBED Equation.DSMT4  after a rotation. Using a compass and straight edge, locate the center of the rotation. (Note: If a rotation carries A onto A', then center of rotation must be on the line of reflection that would carry A onto A'.)  6. The points A and A' have coordinates ((5, 3) and (1, 3) respectively. Under which transformations will A' be the image of A? a. A reflection over the line y = x. b. A 90( counterclockwise rotation about the point (3, 1). c. A translation 4 units left. d. A dilation of scale factor 2 centered at the point (7, 3). e. A reflection over the line x = 3. f. A 180( rotation about the point (3, 3). g. A dilation of scale factor 1/3 centered at the point (1, 3). 7. The image of point A(4, 7) after a 180( rotation around point P is A'(8, 1). What are the coordinates of point P? 8. A translation moves P(5, 2) to P'(1, 3). What is the image of (2, 3) under the same translation? 9. If the image of point A(3, 5) after a reflection in line k is A'(5, 3), find the equation of line k. 10. Under the translation (x, y) ( (x + 3, y 5), what is the pre-image of the point (1, 1)? 11. The vertices of (JKL have coordinates J(5, 1), K(2, 3) and L(4, 1). Under which transformation is the image (J'K'L' not congruent to (JKL? (1) a translation of two units to the right and two units down (2) a counterclockwise rotation of 180 degrees around the origin (3) a reflection over the x-axis (4) a dilation with a scale factor of 2 and centered at the origin 12. A regular polygon has a minimum rotation of 40 to carry the polygon onto itself. a. How many sides does the polygon have? b. What other (positive) rotations will carry this polygon onto itself?  13. In the diagram at right, (OCD is the image of (OAB after a dilation centered at the origin. The coordinates of the vertices are O(0, 0), A(0, 4), B(3, 0), C(0, 6), and D(4.5, 0). The ratio of AB to CD is (1)  EMBED Equation.DSMT4  (3)  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  14. The line  EMBED Equation.DSMT4  is transformed by a dilation centered at the point (0, 3). Which linear equation could be its image? (1)  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4  (3)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  15. The image of (ABC after a dilation of scale factor k centered at point A is (ADE, as shown in the diagram below. Which statement is not always true? (1) 2AB = AD (3) AC = CE (2)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  16. Triangle GRL is the image of (BOY after a dilation of scale factor 2. If BO = x + 3 and GR = 3x 1, then the length of GR is (1) 5 (2) 7 (3) 10 (4) 20 17. Line segment  EMBED Equation.DSMT4  , whose endpoints are (4, 2) and (10, 2), is rotated 90( clockwise about the origin, then dilated by a scale factor of  EMBED Equation.DSMT4  centered at the origin, reflected over the y-axis, and finally translated 3 units right and 2 units up. What is the length of the resulting image? (1) 9 (2) 2 (3) 6 (4) 4 18. Which of the following could not be the minimum number of degrees required to rotate a regular polygon onto itself? ( 1) 15 (2) 22.5 (3) 32 (4) 40  19. Which is the image of under the transformation rx-axis o R90 ? (1) (2) (3) (4) 20. Line ( is mapped onto line m by a dilation centered at the origin with a scale factor of 5. The equation of line ( is 3x 2y = 8. Write an equation for line m. 21. Triangle ABC is graphed at right. Graph (A'B'C', the image of (ABC after a 90(rotation about point P followed by a reflection over line l. 22. The point A is translated right 4 units and up 3 units; call the image B. a. What is the slope of  EMBED Equation.DSMT4 ? b. What is the length of EMBED Equation.DSMT4 ? Segment AB is dilated by a scale factor of 6 centered on point C which is not on  EMBED Equation.DSMT4 . c. What is the slope of the image  EMBED Equation.DSMT4 ? d. What is the length of the image  EMBED Equation.DSMT4 ? 23. In the diagram (which is NOT to scale), (PIG is the image of (COW after a dilation of scale factor k with center S. a. What is the scale factor of the dilation? b. What is the length of  EMBED Equation.DSMT4 ? c. What is the length of  EMBED Equation.DSMT4 ? 24. A square is graphed at right. a. Write equations for all the line of symmetry for the square. b. Describe the squares rotational symmetries. 25. Triangle ABC has perimeter 42 and area 84; (A'B'C' is the image of (ABC after the transformation  EMBED Equation.DSMT4 . a. What is the perimeter of (A'B'C'? b. What is the area of (A'B'C'?  26. The letter R is graphed at the origin. Describe a sequence of transformations that would take the original R into each of the three transformed Rs labeled a, b and c. 27. Given that IBEX ( TAHR, describe a sequence of transformations that will take IBEX onto TAHR.  Stuff you should know: Properties of transformations Orientation Isometries Direct and opposite Preserved properties (length, angle measure, orientation, parallelism, etc.) Stuff you should know: Vocabulary: Transformation Reflection, line of reflection Rotation, center of rotation, angle of rotation Translation, vector Dilation, center of dilation, constant of dilation Image and pre-image Isometry/Rigid motion Orientation; direct and opposite isometries Symmetry (line and rotation) Composition Invariant/preserved Fixed points Congruent Regular polygon Be able to: Reflect a figure over a given line Rotate a figure a given angle about a given point Translate a figure along a given vector Dilate a figure by a given scale factor from a given center Evaluate a composition of transformations State a transformation or composition of transformations that will take a given figure onto its image Identify symmetries of figures Identify fixed (invariant) points of transformations State properties of transformations that are invariant (preserved) Find perimeters and areas after dilations Remember, there are other possible correct answers. K A Y Y' K' A' k S R  x y A B C  x y C A T C' T' A'  x y C A T C' A' T'  x y C A T C' A' T' A C B B' C' A EMBED Equation.DSMT4 ' A C B A' C' B' A B C A' C' B' A' A B C' C' B'  k ( n  y x A B P ( A A' B C A A' B B' C C' A A' B B' C C A B B C C' A A=B' B=A' B' C C' C'    R A T R' A' T' P 60( 120( A three leafed rose has 120( rotational symmetry b/c it is its own image after a 120( rotation. It also has rotational symmetry of all multiples of 120(: 240(. Because it does not have 180( rotational symmetry, it does not have point symmetry. A regular hexagon has 60( rotational symmetry b/c it is its own image after a 60( rotation. It also has rotational symmetry of all multiples of 60(: 120(, 180(, 240( and 300(. Because it has 180( rotational symmetry, it is also said to have point symmetry. A B C P Q A B R S  x y A B C P Q 9. a. Graph DICE having vertices I( 3, 1), C( 1, 0), and E( 1, 4) b. Graph DI'C'E', the image of DICE under a line reflection in the y-axis. c. Graph DI"C"E", the image of DI'C'E' after a line reflection in the line x = 3. d. Name the single transformation that is equivalent to ry-axis followed by rx = 3.   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Mdrs/downrev.xmlLN0HHܨShƩPVK5^Gۆ>= \Vhr5Nic$A\;rc01)"B yKRڐE?q=q>`1D94Rx嶓YØ4 p;fɑ3׶Vc.ӵn(&poPK!8! _rels/.relsj0 }qN/k؊c[F232zQLZ%R6zPT]( LJ[ۑ̱j,Z˫fLV:*f"N.]m@= 7LuP[i?T;GI4Ew=}3b9`5YCƵkρؖ9#ۄo~e?zrPK!\4+drs/e2oDoc.xmlSn0 }q65]  YI&)%'i Ӄ@]xsHnA2:H+Qvͻ%%!rp V2"]}]-+@7]. :ix[G<]x#EUWEq oGm+EڶAFb}޷i/+^BZox(At˹6)z^]-(6"?J0$z,~S:>}I,lֹ%=7j.^؟ZFeZc$l#Wz1Gډ9aTx([h^P cCсEIh^R?٤Z |@y؞ n3)< ިkcXYν<_ PK! Mdrs/downrev.xmlLN0HHܨShƩPVK5^Gۆ>= \Vhr5Nic$A\;rc01)"B yKRڐE?q=q>`1D94Rx嶓YØ4 p;fɑ3׶Vc.ӵn(&poPK!8! _rels/.relsj0 }qN/k؊c[F232zQLZ%R6zPT]( LJ[ۑ̱j,Z˫fLV:*f"N.]m@= 7LuP[i?T;GI4Ew=}3b9`5YCƵkρؖ9#ۄo~e?zrPK!\4+drs/e2oDoc.xmlSn0 }q65]  YI&)%'i Ӄ@]xsHnA2:H+Qvͻ%%!rp V2"]}]-+@7]. :ix[G<]x#EUWEq oGm+EڶAFb}޷i/+^BZox(At˹6)z^]-(6"?J0$z,~S:>}I,lֹ%=7j.^؟ZFeZc$l#Wz1Gډ9aTx([h^P cCсEIh^R?٤Z |@y؞ n3)< ިkcXYν<_ PK! Mdrs/downrev.xmlLN0HHܨShƩPVK5^Gۆ>= \Vhr5Nic$A\;rc01)"B yKRڐE?q=q>`1D94Rx嶓Y9 _S N ^`pG?"Text Box 2s"{?PK!8[Content_Types].xmlAN0EH%N@%邴K@`dOdlyLhoDX3'AL:*/@X*eRp208J妾)G,R}Q)=HiҺ0BL):T뢸WQDY;d]6O&8* VCLj"󃒝 yJ.;[wIC_ :{IOA !>Ø4 p;fɑ3׶Vc.ӵn(&poPK!8! _rels/.relsj0 }qN/k؊c[F232zQLZ%R6zPT]( LJ[ۑ̱j,Z˫fLV:*f"N.]m@= 7LuP[i?T;GI4Ew=}3b9`5YCƵkρؖ9#ۄo~e?zrPK!\4+drs/e2oDoc.xmlSn0 }q65]  YI&)%'i Ӄ@]xsHnA2:H+Qvͻ%%!rp V2"]}]-+@7]. :ix[G<]x#EUWEq oGm+EڶAFb}޷i/+^BZox(At˹6)z^]-(6"?J0$z,~S:>}I,lֹ%=7j.^؟ZFeZc$l#Wz1Gډ9aTx([h^P cCсEIh^R?٤Z |@y؞ n3)< ިkcXYν<_ PK! Mdrs/downrev.xmlLN0HHܨShƩPVK5^Gۆ>= \Vhr5Nic$A\;rc01)"B yKRڐE?q=q>`1D94Rx嶓YØ4 p;fɑ3׶Vc.ӵn(&poPK!8! _rels/.relsj0 }qN/k؊c[F232zQLZ%R6zPT]( LJ[ۑ̱j,Z˫fLV:*f"N.]m@= 7LuP[i?T;GI4Ew=}3b9`5YCƵkρؖ9#ۄo~e?zrPK!\4+drs/e2oDoc.xmlSn0 }q65]  YI&)%'i Ӄ@]xsHnA2:H+Qvͻ%%!rp V2"]}]-+@7]. :ix[G<]x#EUWEq oGm+EڶAFb}޷i/+^BZox(At˹6)z^]-(6"?J0$z,~S:>}I,lֹ%=7j.^؟ZFeZc$l#Wz1Gډ9aTx([h^P cCсEIh^R?٤Z |@y؞ n3)< ިkcXYν<_ PK! 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