Transformations. Transformations: CLASSWORK. Tell whether the transformation appears to be a rigid motion. Explain

Transcription

1 Transformations Transformations: CLASSWORK Tell whether the transformation appears to be a rigid motion. Explain Preimage Image Preimage Image 3. Identify the type of transformation. What is the image of segment FG? 4. Identify the type of transformation. What is the image of angle J? Geometry Transformations ~1~ NJCTL.org

2 Transformations: HOMEWORK Tell whether the transformation appears to be a rigid motion. Explain Preimage Image Preimage Image 3. Identify the type of transformation. What is the image of segment QR? 4. Identify the type of transformation. What is the image of angle T? Geometry Transformations ~2~ NJCTL.org

3 Translations: CLASSWORK Graph the image of each figure under the given translation. 1. T< 1, 4> (ΔABC) 2. T<3, -3> (JKLM) 3. Write a rule to describe the translation. 4. If T<10, 7> (QRST) = Q R S T, what translation maps Q R S T onto QRST? Geometry Transformations ~3~ NJCTL.org

4 5. ΔXYZ has coordinates X(2, 3), Y(1, 4), and Z(8, 9). A translation maps X to X (4, 8). What are the coordinates for Y and Z for this translation? 6. Write three different translation rules for which the image of ΔABC has a vertex at the origin. Translations: Homework Graph the image of each figure under the given translation. 1. T<3, -2> (ΔDEF) 2. T<-4, 0> (WXYZ) Geometry Transformations ~4~ NJCTL.org

5 3. Write a rule to describe the translation. 4. If T<-4, 9> (ΔLMN) = ΔL M N, what translation maps ΔL M N onto ΔLMN? 5. ΔABC has coordinates A(2, 3), B(4, 2), and C(3, 0). After a translation the coordinates of A are (6, 2). What are the coordinates of B and C? 6. Write three different translation rules for which the image of ΔBCD has a vertex at the origin. Geometry Transformations ~5~ NJCTL.org

6 Reflections: CLASSWORK For numbers 1-6 find the coordinates of each image. 1. Rx-axis (A) 2. Ry-axis (B) 3. Ry = x (C) 4. Rx = 2 (D) 5. Ry= -1 (E) 6. Rx = -3 (F) 7. Given points A(3, 3), B(5, -2), and C(4, 4), graph ΔABC and its reflection image as indicated. a. Ry-axis (ΔABC) b. Rx-axis (ΔABC) Geometry Transformations ~6~ NJCTL.org

7 c. Rx = 1(ΔABC) d. Ry = 2(ΔABC) Draw the line of reflection you can use to map one figure onto the other Find the image of J(-2, 1) after two reflections, first across line l1, and then across line l2. a. l 1 : x = 2, l 2 : y-axis b. l 1 : y = 1, l 2 : y-axis c. l 1 : y = 2, l 2 : x-axis d. l 1 : x = 1, l 2 : y = Give an example of a place you may see a geometric reflection in everyday life. Explain. Geometry Transformations ~7~ NJCTL.org

8 Identify all of the lines of symmetry for the given shape Reflections: HOMEWORK For numbers 1-6 find the coordinates of each image. 1. Rx-axis (A) 2. Ry-axis (B) 3. Ry = x (C) 4. Rx = 3 (D) 5. Ry= 2 (E) 6. Rx = -1 (F) Geometry Transformations ~8~ NJCTL.org

9 7. Given points D(-2, 1), E(1, 3), and F(2, -2), graph ΔDEF and its reflection image as indicated. a. Ry-axis (ΔDEF) b. Rx-axis (ΔDEF) c. Rx = 2(ΔDEF) d. Ry = 1(ΔDEF) Geometry Transformations ~9~ NJCTL.org

10 Draw the line of reflection you can use to map one figure onto the other Find the image of K(4, -3) after two reflections, first across line l1, and then across line l2. a. l 1 : x = 2, l 2 : y-axis b. l 1 : y = 1, l 2 : y-axis c. l 1 : y = 2, l 2 : x-axis d. l 1 : x = 1, l 2 : y = Give an example of a place you may see a geometric reflection in everyday life (Different than your classwork answer). Explain Identify all of the lines of symmetry for the given shape Geometry Transformations ~10~ NJCTL.org

11 Rotations: CLASSWORK Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image. 1. r(-60, P)( ABC) 2. r(85, P)( ABC) 3. Point O is the center of regular pentagon JKLMN. Find the image of the given point or segment for the given rotation. (counterclockwise) a. r(144, O)(K) b. r(72, O)(N) c. r(216, O)(ML) d. r(360, O)(JN) e. r(288, O)(JO) Geometry Transformations ~11~ NJCTL.org

12 4. ΔABC has vertices A(2, 2), B(3, 2), and C( 1, 3). a. Graph r(270º, O)( ABC). b. Graph r(180º, O)( ABC). c. Graph r(90º, O)( ABC). 5. The vertices of ABCD have coordinates A(2, 6), B(-3, 4), C(-3, -5), and D(4, -2). What are the coordinates of the vertices of r (180º, O)(ABCD)? Geometry Transformations ~12~ NJCTL.org

13 6. The vertices of r(270º, O) (DEFG) have coordinates D (4, 5), E (4, -3), F (-2, -3), and G (-2, 5). What are the coordinates of the vertices of DEFG? 7. ABCD has vertices A(4, 2), B( 2, 2), C( 4, 2), and D(2, 2). Which of the following quadrilaterals is r (180º, O)(ABCD)? a. ABCD b. BCDA c. CDAB d. DABC 8. FGH has vertices F( 1, 2), G(0, 0), and H(3, 1). What are the coordinates of the vertices of r( 270º, G)( FGH)? Do the following figures have rotational symmetry? If yes, what is the degree of rotation? Rotations: HOMEWORK Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image. 1. r(-90, P)( ABC) Geometry Transformations ~13~ NJCTL.org

14 2. r(110, P)( ABC) 3. Point O is the center of regular hexagon ABCDEF. Find the image of the given point or segment for the given rotation. (counterclockwise) a. r(120, O)(F) b. r(180, O)(B) c. r(300, O)(BC) d. r(360, O)(FE) e. r(60, O)(E) f. r(240, O)(AB) 4. Quadrilateral DEFG has vertices D(2, 3), E(4, -2), F(7, -2), and G(5, 2). a. Graph r(270º, O)(DEFG). b. Graph r(180º, O)(DEFG). Geometry Transformations ~14~ NJCTL.org

15 c. Graph r(90º, O)( DEFG). 5. The vertices of PQRS have coordinates P( 1, 5), Q(3, 4), R(2, 4), and S( 3, 2). What are the coordinates of the vertices of r (90º, O)(PQRS)? 6. The vertices of r(270º, O) (KLMN) have coordinates K ( 3, 2), L (2, 3), M (4, 2), and N ( 2, 4). What are the coordinates of the vertices of KLMN? 7. The vertices of quadrilateral ABCD have coordinates A(4, 3), B( 3, 4), C( 4, 3), and D(3, 4). Explain how the transformation r(90º, O)(ABCD) = BCDA can be used to show that the quadrilateral is square. Geometry Transformations ~15~ NJCTL.org

16 8. RST has vertices at R(0, 3), S(4, 0), and T(0, 0). What are the coordinates of the vertices of r(- 270º,T)( RST)? Do the following figures have rotational symmetry? If yes, what is the degree of rotation? Geometry Transformations ~16~ NJCTL.org

17 Compositions of Transformations: CLASSWORK Find the image of each letter after the transformation Rm º Rl. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation. l 1. l 2. J 60º m K m P For numbers 3 6 Graph ABC and its glide reflection image. A(-3, 3), B(0, 1) and C(-1,-4) 3. (Ry-axis T<3, 0>)( ABC) 4.(Ry=1 T< 1, 0>)( ABC) Geometry Transformations ~17~ NJCTL.org

18 5. (Rx=2 T<0, 1>)( ABC) 6. (Ry=x T<3, 3>)( ABC) 7. Lines l and m intersect at point P and are perpendicular. If a point Q is reflected across l and then across m, what transformation rule describes this composition? 8. Graph AB and its image A B after a reflection first across l1 and then across l2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation. A( 3, 4), B( 1, 0); l1 : x = 1; l2 : y = 1 Geometry Transformations ~18~ NJCTL.org

19 9. P maps to P (2, 3) by the given glide reflection. What are the coordinates of P? (Rx-axis º T <2, 1>)(P) Compositions of Transformations: HOMEWORK Find the image of each letter after the transformation Rm º Rl. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation. m A l K N l m For numbers 3 6 Graph DEF and its glide reflection image. D(0, 2), E(2, 4) and F(4,0) 3. (Rx-axis T<2, 1>)( DEF) 4. (Ry=2 T< 3, 1>)( DEF) Geometry Transformations ~19~ NJCTL.org

20 5. (Rx=1 T<0, -2>)( DEF) 6.. (Ry=x T<2, 3>)( DEF) 7. A triangle is reflected across line l and then across line m. If this composition of reflections is a translation, what is true about m and l? 8. Graph AB and its image A B after a reflection first across l1 and then across l2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation. A( 5, 2), B( 3, 6); l1 : x = 2; l2 : x = 3 Geometry Transformations ~20~ NJCTL.org

21 9. P maps to P (2, 3) by the given glide reflection. What are the coordinates of P? (Ry = x º T <3, 3>)(P) Congruence Transformations: CLASSWORK ABC DEF. What is a congruence transformation that maps ABC to DEF? Identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image Geometry Transformations ~21~ NJCTL.org

22 Congruence Transformations: HOMEWORK ABC DEF. What is a congruence transformation that maps ABC to DEF? Identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image Geometry Transformations ~22~ NJCTL.org

23 Dilations: CLASSWORK A dilation has center (0, 0). Find the image of each point for the given scale factor. 1. A(3, 4); D7(A) 2. B(0, 4); D3.4(B) 3. C(5, 6); D5/3(C) 4. A square has 16-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor Graph quadrilateral ABCD and its image A B C D for a dilation with center (0, 0) and a scale factor of 2. The vertices of ABCD are: A(2, 3), B(-2, 4), C(-3, -2), D(3, 3). Geometry Transformations ~23~ NJCTL.org

24 The solid-line figure is a dilation of the dashed-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation D(2, K)( ABC) 9. ABC is dilated in the coordinate plane with center (2, -1) and scale factor 0.5. If A(6, 3) what is the y-coordinate of A? Dilations: HOMEWORK A dilation has center (0, 0). Find the image of each point for the given scale factor. 1. P( 3, 5); D1. 2(P) 2. O( 2, 1); D4(O) Geometry Transformations ~24~ NJCTL.org

25 3. M(4, 2); D3/2(M) 4. A square has 12-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor Graph pentagon JKLMN and its image J K L M N for a dilation with center (0, 0) and a scale factor of 1.5. The vertices of JKLMN are: J(0, 3), K(3, 3), L(3, 0), M(0, 3), N( 1, 0). Geometry Transformations ~25~ NJCTL.org

26 The dashed-line figure is a dilation of the solid-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation D(3, A)( ABC) 9. DEFis dilated in the coordinate plane with center (-5, 2) and scale factor 2. If B(-5, 4) what is the y-coordinate of B? Geometry Transformations ~26~ NJCTL.org

27 Similarity Transformations: CLASSWORK JKM has vertices J(0, 4), K(4, 4) and M(0, 3). Draw the image for each similarity transformation. 1. (D2 Rx-axis) ( JKM) 2. (T<-3, -2> D1.5 ) ( JKM) 3. (D0.5 r(270º, O)) ( JKM) Geometry Transformations ~27~ NJCTL.org

28 4. For the graph, describe the composition of transformations that maps ΔDEF to ΔLMN. 5. Show that ΔABC~ΔFGH by finding a similarity transformation that maps one triangle to the other. 6. Determine if there is a similarity transformation that maps one figure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain. 7. Do the compositions T<4, 2> D2 and D2 T<2, 1> describe the same similarity transformation? Explain. Geometry Transformations ~28~ NJCTL.org

29 Similarity Transformations: HOMEWORK JKM has vertices J(-2, 2), K(2, 0) and M(1, -2). Draw the image for each similarity transformation. 1. (D2 Ry-axis) ( JKM) 2. (D0.5 T<2, -2>) ( JKM) 3. (r(-90º, O) D1.5 )( JKM) Geometry Transformations ~29~ NJCTL.org

30 4. For the graph, describe the composition of transformations that maps ΔDEF to ΔLMN. 5. Show that ΔABC~ΔFGH by finding a similarity transformation that maps one triangle to the other. 6. Determine if there is a similarity transformation that maps one figure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain. 7. A similarity transformation is described by the composition T<2, 1> D3. If the order of the composition is changed to be D3 T<2, 1>,does that describe the same transformation? Explain. Geometry Transformations ~30~ NJCTL.org

31 ANSWER KEY Transformations: CLASSWORK 1. Yes, distances and angle measures are preserved. 2. No, distances are not preserved. 3. Rotation; F G 4. Dilation; angle J Transformations: HOMEWORK 1. No, distances are not preserved. 2. Yes, distances and angle measures are preserved. 3. Reflection; segment G R 4.Translation; angle T Translations: CLASSWORK Geometry Transformations ~31~ NJCTL.org

32 3. T<10, -1> (ABDC) 4. T<-10, -7> (Q R S T ) = QRST 5. Y (3, 9); Z (10, 14) 6. T<-2, -2> (ΔABC); T<-2, 6> (ΔABC); T<-5, -3> (ΔABC) Translations: HOMEWORK Geometry Transformations ~32~ NJCTL.org

33 3. T<-4, 10> (ΔIJK) 4.T<4, -9> (ΔL M N ) = ΔLMN 5. B (8, -7); C (7, -5) 6. T<-4, -2> (ΔBCD); T<-4, -9> (ΔBCD); T<-8, -6> (ΔBCD) Reflections: CLASSWORK 1. A (1, -3) 2. B (-5, 5) 3. C (-1, 3) 4. D (7, 1) 5. E (-2, 1) 6. F (-3, 4) 7. a. A (-3, 3), B (-5, -2), C (-4, 4) b. A (3, -3), B (5, 2), C (4, -4) c. A (-1, 3), B (-3, -2), C (-2, 4) d. A (3, -7), B (5, -2), C (4, -8) Geometry Transformations ~33~ NJCTL.org

34 a. J (-6, 1) b. J (2, -3) c. J (-2, 1) d. J (0, -7) 11. Answers may vary. Sample answer: Patterns in tiles, wallpaper, etc. Reflections: HOMEWORK 1. A (5, -1) 2. B (-3, 2) Geometry Transformations ~34~ NJCTL.org

35 3. C (-2, 1) 4. D (12, 1) 5. E (-1, 4) 6. F (0, -5) 7. a. D (2, 1), E (-1, 3), F (-2, -2) b. D (-2, -1), E (1, -3), F (2, 2) c. D (6, 1), E (3, 3), F (2, -2) d. D (-2, -3), E (1, -5), F (2, 0) Geometry Transformations ~35~ NJCTL.org

36 10. a. K (0, -3) b. K (-4, 1) c. K (4, -7) d. K (-6, -3) 11. Answers may vary. Sample answer: Patterns in tiles, wallpaper, etc. Rotations: CLASSWORK Geometry Transformations ~36~ NJCTL.org

37 3. a. N b. M c. NJ d. JN e. KO 4. a. A (-2, 2), B (2, 3), C (-3, -1) b. A (-2, -2), B (-3, 2), C (1, -3) c. A (2, -2), B (-2, -3), C (3, 1) 5. A (-2, 6), B (3, -4), C (3, 5), D (-4, 2) 6. D (5, -4), E (-3, -4), F (-3, 2), G (5, 2) 7. C Geometry Transformations ~37~ NJCTL.org

38 8. F (-2, -1), G (0, 0), H (1, 3) 9. Yes, 180º 10. Yes, 90º 11. Yes, 180º Rotations: HOMEWORK a. D b. E c. DC d. FE e. D f. DC 4. a. D (-3, 2), E (2, 4), F (2, 7), G (-2, 5) b. D (-2, -3), E (-4, 2), F (-7, 2),G (-5, -2) Geometry Transformations ~38~ NJCTL.org

39 c. D (3, -2), E (-2, -4), F (-2, -7),G (2, -5) 5. P (5, 1), Q (4, -3), R (-4, -2), S (-2, 3) 6. K(2, 3), L(3, -2), M(-2, -4), N(-4, 2) 7. Answers may vary. Sample: A rotation is a rigid motion therefore distances and angle measures are preserved. AB = BC = CD = AD and m<a = m<b = m<c = m<d. If all angles of a quadrilateral are equal then they each must measure 90 degrees. By definition the quadrilateral is a square. 8. R (-3, 0), S (0, 4), T (0, 0) 9. Yes, 45º 10. Yes, 22.5º 11. Yes, 180º Compositions of Transformations: CLASSWORK 1. translation, down and to the right, twice the distance between lines l and m 2. rotation, about point P, 120º clockwise Geometry Transformations ~39~ NJCTL.org

40 3. A (0, 3), B (-3, 1), C (-2, -4) 4. A (-4, -1), B (-1, 1), C (-2, 6) 5. A (7, 4), B (4, 2), C (5, -3) 6. A (6, 0), B (4, 3), C (-1, 2) º rotation 8. A rotation, about point (1, -1), 180º clockwise Geometry Transformations ~40~ NJCTL.org

41 9. P(0, -4) Compositions of Transformations: HOMEWORK 1. rotation, about point K, 180º counterclockwise 2. translation, directly down, twice the distance between lines l and m 3. D (2, -3), E (4, -5), F (6, -1) 4. D (-3, 1), E (-1, -1), F (1, 3) 5. D (2, 0), E (0, 2), F (-2, -2) 6. D (5, 2), E (7, 4), F (3, 6) Geometry Transformations ~41~ NJCTL.org

42 7. m l 8. A translation, 10 units to the right 9. P(0, -1) Congruence Transformations: CLASSWORK Geometry Transformations ~42~ NJCTL.org

43 1. Check student work. Sample Answer: (T <1, 0>º Ry -axis º r(270º, O))(ΔABC) = ΔDEF 2. Check student work. Sample Answer: (T <2, 0>º Ry -axis º r(180º, O))(ΔABC) = ΔDEF 3. ΔABC ΔGHI; Check student work. Sample Answer: r(180º, O))(ΔABC) = ΔGHI 4. ABCD LIJK; Check student work. Sample Answer: T <-5, -4>(ABCD) LIJK Congruence Transformations: HOMEWORK 1. Check student work. Sample Answer: (Rx -axis º T <5, 0> (ΔABC) = ΔDEF 2. Check student work. Sample Answer: (r(90º, O) º Ry -axis )(ΔABC) = ΔDEF 3. CD EF; Check student work. Sample Answer: Ry -axis (CD) = EF 4. ΔABC ΔIHG; Check student work. Sample Answer: Ry =2 ºT <6, 0> (ΔABC) = ΔIHG Dilations: CLASSWORK 1. A (21, 28) 2. B (0, 13.6) 3. C (25/3, -10) 4. A square with 12.8 cm sides. 5. A (4, 6), B (-4, 8), C (-6, -4), D (6, -6) 6. scale factor: 3; enlargement 7. scale factor: 1/4; reduction 8. Geometry Transformations ~43~ NJCTL.org

44 9. 1 Dilations: HOMEWORK 1. A (-3.6, 6.0) 2. B (-8, -4) 3. C (6, 3) 4. A square with 9.6 cm sides. 5. J (0, 4.5), K (4.5, 4.5), L (4.5, 0), M (0, -4.5), N (-1.5, 0) 6. scale factor: 9/5; enlargement 7. scale factor: 1/3; reduction Geometry Transformations ~44~ NJCTL.org

45 Similarity Transformations: CLASSWORK 1. J (0, -8), K (8, -8), M (0,6) 2. J (-3, 4), K (3, 4), M (-3, 2.5) Geometry Transformations ~45~ NJCTL.org

46 3. J (-2, 0), K (-2, 2), M (-1.5, 0) 4. Answers may vary. Sample: (D0.5 ºr(180,O))(ΔDEF) = ΔLMN 5. Answers may vary. Sample: (T<-4, -10>º D3)(ΔABC) = ΔFGH 6. No, the corresponding sides are not proportional. 7. Yes. For a point (x, y), the first transformation will transform (x, y) to (2x + 4, 2y-2) while the second transformation will transform it to (2(x+2), 2(y-1)) which = (2x + 4, 2y-2) Similarity Transformations: HOMEWORK 1. J (4, 4), K (-4, 0), M (-2, -4) Geometry Transformations ~46~ NJCTL.org

47 2. J (0, 0), K (2, -1), M (1.5, -2) 3. J (-3, -3), K (0, 3), M (3, 1.5) 4. Answers may vary. Sample: (Rx -axis º D2)(ΔDEF) = ΔLMN 5. Answers may vary. Sample: (r(90,o) º D1.5 )(ΔABC) = ΔFGH 6. Yes. A reflection followed by a dilation with a scale factor of 0.5. ΔABC~ΔEDC 7. No. For a point (x, y), the first transformation will transform (x, y) to (3x + 2, 3y+1) while the second transformation will transform it to (3(x+2), 3(y+1)). Geometry Transformations ~47~ NJCTL.org