Rigid Motions HW packet. Name:
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1 Rigid Motions HW packet Name:
2 Compositions of Transformations The transformations you have learned so far are: Name Meaning Notation Translations add the specified amount to the coordinates of each point. T3,4 Reflect in x change the sign of the y-coordinate. rx-axis Reflect in y change the sign of the x-coordinate ry-axis Reflect in y=x switch the coordinates of each point ry=x Reflect in y=-x switch and negate the coordinates of each point. ry=-x Rotate 90 o switch the order and then change the sign of the x-coordinate. R90 Rotate 180 o pply the 90 o rotation twice R180 Rotate 270 o pply the 90 o rotation three times R270 Use the rules to solve the problems below. 1. (-3, 5) T (2,1) 2. (-2, -4) T (2,1) 3. C(3, -5) r xaxis 4. D(5, 3) r yaxis 5. E(-3, -2) r yx 6. R(6, 2) R S(-5, 3) R T(7, -1) R r yx F (7, -2) 10. r yaxis G (3, -9) T (3,5 ) 11. J (7, 3) 12. r xaxis K (-3, 2)
3 Page 3 of 18 shape can be transformed more than once. Such a transformation is called a composition of transformations. The transformations are written together with a small between them. r x T 2 1, 5 means translate first and then reflect. T r ( means first and then. 2, 4) yaxis r y axis R360 R means first and then. 90 T 3, 2 means first and then. Now answer the following. Show all your work. R? 1. What is the image of (-3, 5) under 90 T 1, 5 T r? 2. What is the image of P(-3, -9) under ( 2, 4) y1 3. What is the image of S(3, 6) under R 180 T (1, 2)? 4. What is the image of Q(-2, 8) under ry axis ry 1? 5. Graph C with (3, 5), (5,3) and C(3, -1). Graph the image of C under r x axis T 5, 3 6. Graph PQR with P(2, -1), Q(2,1) and R(-2, 0). Graph the image of PQR under r axis T 4,4 y.
4 Page 4 of Graph DEF with D(-5, 5), E(-8,2) and F(-3, 0). Graph the image of DEF under T 2,1 R Graph DEF with D(4,4), E(6,2) and F(2,0). Graph the image of DEF under r x axis R Graph C with (8, 1), (6,3) and C(5, -1). Graph the image of C under R 180 T 3, Graph DEF with D(-1,-5), E(-5,-4) and F(-3,-1). Graph the image of DEF under T 3,2 R 90.
5 Page 5 of Graph DEF with D(1,1), E(5,1) and F(3,3). Graph the image of DEF under r yx R Graph DEF with D(1,-1), E(3,-1) and r r. F(2,-4). Graph the image of DEF under yaxis x3 12. Graph C with (-1,1), (-4,3) and C(-2, 0). Graph the image of C under R T ,0 14. Graph DEF with D(3,3), E(9,6) and T. F(6,9). Graph the image of DEF under, R
6 Page 6 of 18 Transformation Figure re Corresponding Parts Congruent? (is it a rigid motion?) Segments between Corresponding Points Other Relationships Coordinate Rules Rotation Reflection
7 Page 7 of 18 Transform - ation Figure re Corresponding Parts Congruent? (is it a rigid motion?) Segments between Corresponding Points Congruent? Other Relationships Coordinate Rules Translation Dilation Mapping a regular polygon to itself:
8 Page 8 of 18 Transformations Practice #1 Multiple Choice 1. Which of the following transformations always preserves slope? (1) translation (3) reflection (2) rotation (4) glide reflection 2. In the accompanying figure, ''C' is the image of C after a translation. Which of the following is not necessarily true? (1) (3) (2) // (4) 3. In the accompanying figure, DFE is the image after RO, 80( C). Which if the following must be true? C F O (1) m 80 (3) m m C (2) mof = 80 (4) me = 80 D E 4. is the reflection of over line m. Which if the following is not necessarily true? (1) the midpoint of GE lies on line m (2) GE = OW (3) line m (4) O G E m W 5. MNP is the reflection of ZXY through point O. Which of the following must be true? (1) YM = XN (2) YZ = ZX (3) MN = YZ (4) YZ = MP X Y M O Z N P
9 Page 9 of Pentagon CDE is the image of pentagon VWXYZ after DC,3. Which of the following must be true? (1) VW = 3 (3) CV = 3V (2) CV = 3C (4) m = 3(mV) 7. Given RO,110( ) =. If mhog = 45, what is the measure of GOP? H O Q (1) 25 (2) 45 (3) 65 (4) 70 G P 8. Which transformation maps C onto ''C'? C ' (1) R,90 (2) R,90 (3) (4) C' 9. Given the following angle measures, which point could be the center of rotation that maps C onto ''C'? m X' = 30 o m Y' = 40 o C' C ' m CXC' = 30 m CYC' = 35 o o m Z' = 45 m CZC' = 48 o o Z X Y ' (1) X (2) Y (3) Z (4) neither X, Y, or Z could be the center 10. In regular hexagon CDEF, which transformations would map QF to DQC? F E (1) (2) reflect through point Q (3) (4) RQ,60 Q C D
10 Page 10 of In the figure to the right RPS measures 30, and is rotated 160about point P. R'PS' is the image after the rotation. Find the measure of SPR'. 12. Describe fully the transformation that maps XYZ to XY'Z'. 13. When an equilateral triangle is reflected over one of its sides, what type of figure is formed? Sketch an example. 14. Given point Y is the image of point X after a reflection over, explain why PXZ PYZ? Z X P Y 15. Estimate the location of the center of rotation. Use a protractor to determine how close your estimate was by measuring the angle formed by two pairs of corresponding points and your center. ' C' ' D' E' F' C D E F
11 Page 11 of Sketch segment and midpoint P on. Sketch the rotation of about point P by 90. Repeat for point C that lies on but is not the midpoint. 17. Sketch the line of reflection of the following figure: T T 18. Sketch (P) P 21. Sketch the dilation of the triangle through point O with a scale factor of 2. O 22. Shaina states that whenever a segment is reflected over a line, the two endpoints of the image and the two endpoints of a pre-image form a trapezoid. Is she correct? Justify your reasoning.
12 Transformations Practice#2 Page 12 of Which letter has rotational symmetry? (1) T (2) (3) Z (d ) E 2. What is the smallest angle of rotation that would map a hexagon onto itself? (1) 45 (2) 60 (3) 120 (4) Which of the following figures has rotational symmetry? (1) equilateral triangle (2) scalene triangle (3) obtuse triangle (4) right triangle 4. Which of the following figures have exactly two lines of symmetry? (1) 2) 3) 4) 5. y how many degrees must the octagon be rotated to map point X to point Y? X Y (1) 45 (2) 90 (3) 135 (4) y how many degrees must the decagon be rotated to map point R to point S? R (1) 36 (2) 144 (3) 216 (4) 324 S
13 7. Which word has a horizontal line of symmetry? Page 13 of 18 (1) HIDE (2) OSS (3) T (4) TOOT 8. If pentagon GHIJK is rotated 288 about its center P, then vertex G maps to vertex H. What is the measure of the acute angle GPQ? (1) 36 (2) 72 (3) 84 (4) Sketch all lines of symmetry in the following figure. 10. Danny and Sue are riding a Ferris wheel at a carnival. The Ferris wheel has 20 cars and spins at a rate of 6 degrees per seconds. Danny and Sue are in separate cars that are three positions apart. When Danny is at the top of the Ferris wheel, how many seconds later will Sue be at the top? 11. Sketch an example of a figure that has one line of symmetry but does not have rotational symmetry, 12. Jessie states that any triangle with line symmetry must be equilateral, but Fiona disagrees. Who is correct? Justify your answer.
14 Transformations Practice #3 Page 14 of Which composition could result in (12,4) mapping to (5,1)? (1) D1/4 ry=x (3) R90 T9,-11 (2) D1/4 T2,0 (4) T9,-11 R90 2. Which composition of transformations would not fit the two puzzle pieces together D C (1) Tuuuu r orgf G ( HEFG ) (2) Tuuuu r orgf F ( HEFG ) (3) r HE o Tuuu r H ( HEFG ) (4) Tuuuu r orhe H ( HEFG ) H E G F 3. Which composition could result in the following pre-image and image? (1) R 180 o D 1/2 (2) D 2 o R 180 (3) r y-axis o D 1/2 (4) D 1/2 ot 5,0 4. Which composition could result in (5,1) (1,3) (1) T-6,2(rx-axis) (2) rx-axis(t6,2) (3) R90(T2,2) (4) T6,2(ry-axis) 5. Which image represents reflection following by a translation parallel to the line of reflection? (1) (2) (3) (4) 6. Rayquan is designing a website logo using a computer graphics program. He uses the programs enlarge feature to double the logo s size, and then drags it horizontally 2 inches right and vertically 3 inches up. Which composition represents the change in the logo?
15 Page 15 of 18 (1) T 2,3 o D 1/2 (2) T 2,3 o D 2 (3) T 3,2 o D 1/2 (4) D 2 ot 3,2 7. When equilateral triangle C undergoes the transformation R,240 or, which corresponding C pairs of vertices coincide? (1), '' and, '' only (2), '' and C, C'' only (3) C, C'' and, '' only (4), '',, '' and C, C'' 8. The figure represents r line s ( R C,120 ( ) ). Sketch line s and point C in the figure. '' ' ' '' 9. The vertices of IG have coordinates (6, 1) I(3, 2) and G(7, 5). Graph and label ''I''G'', the image of IG after R y=x o R origin,90. State the coordinates. 10. The vertices of PMN have coordinates P(4, 0) M(3, 3) and N(6, 5). Graph and label P''M''N'', the image of PMN after T -4,1 o R x=3. State the coordinates.
16 Congruence with rigid motions Page 16 of In the figure below, D ED EC CFE. Which of the following sequences of rigid motions will always map D to CFE? D E F C (1) a translation by vector E followed by a translation by vector EC (2) a rotation of 180 about point, followed by a translation by vector CF (3) a reflection over E followed by a translation by vector C 2. Triangles MN and PQ share vertex. circle can be constructed that has a center at and passes through points M, N, P, and Q. If mmn = mpq, Which of the following is not necessarily true?? Q P M N (1) NQ is a right angle (2) M Q (3) PQ is the image of MN after a rotation about point. (4) MN and QP are isosceles 3. Regular hexagon CDEF is shown in the figure below. Which of the following transformations will map the hexagon onto itself such that point D maps to point E? C (1) a 60 rotation bout point P followed by a reflection of line E (2) reflection over line E followed by 60 rotation about point P P D (3) reflection over lines CF followed by a translation by vector FE (4) reflection over lines CF followed by a 120 rotation about point P F E
17 4. Which of the following will map parallelogram CD onto itself? D C Page 17 of 18 (1) rotation of 180 about the point of intersection of the two diagonals of the CD (2) reflection over CD, followed by a translation along vector D (3) rotation of 180 about point, followed by a translation along vector (4) reflection over C, followed by a translation a rotation of 180 about point 5. Given right triangles C and DEF where C and F are right angles, C DF and C FE. Describe a precise sequence of rigid motions which would show C DEF. 6.
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