Lesson 9.1 Properties of Transformations

Transcription

1 Lesson 9.1 roperties of Transformations Name eriod Date In Eercises 1 3, draw the image according to the rule and identif the tpe of transformation. 1. (, ) (, ) 2. (, ) ( 4, 6) 3. (, ) (4, ) In Eercises 4 and 5, the Harbour High Geometr Class is holding a Fence ace. Contestants must touch each fence at some point as the run from S to F. Use our geometr tools to draw the shortest possible race path Fence Fence 1 S S F F Fence 2 In Eercises 6 8, complete the ordered pair rule that transforms each triangle to its image. Identif the transformation. Find all missing coordinates. 6. (, ) (, ) 7. (, ) (, ) 8. (, ) (, ) B( 5, 2) C ( 5, 4) (5, 2) B (8, 2) Q( 2, 1) Q (2, 1) ( 7, 3) (4, 5) T(0, 7) S(3, 3) S (3, 0) T 1

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3 9-2 NME DTE EID Translations Translations Using Coordinates transformation called a translation slides a figure in a given direction. In the coordinate plane, a translation moves ever preimage point (, ) to an image point ( a, b) for fied values a and b. In words, a translation shifts a figure a units horizontall and b units verticall; in smbols, (, ) ( a, b). Eample ectangle ECT has vertices ( 2, 1), E( 2, 2), C(3, 2), and T(3, 1). Graph ECT and its image for the translation (, ) ( 2, 1). The translation moves ever point of the preimage right 2 units and down 1 unit. (, ) ( 2, 1) ( 2, 1) ( 2 2, 1 1) or (0, 2) E( 2, 2) E ( 2 2, 2 1) or E (0, 1) C(3, 2) (3 2, 2 1) or (5, 1) T(3, 1) T (3 2, 1 1) or T (5, 2) E E C T T Eercises Graph each figure and its image under the given translation. Lesson Q with endpoints ( 1, 3) and Q(2, 2) under the translation left 2 units and up 1 unit Q Q 2. Q with vertices ( 2, 4), Q( 1, 2), and (2, 1) under the translation right 2 units and down 2 units Q Q 3. square SQU with vertices S(0, 2), Q(3, 1), U(2, 2), and ( 1, 1) under the translation right 3 units and up 1 unit S S U Q U Q 3

4 9-2 NME DTE EID Translations Translations b epeated eflections nother wa to find the image of a translation is to reflect the figure twice in parallel lines. This kind of translation is called a composite of reflections. Eample In the figure, m n.find the translation image of BC. B is the image of BC reflected in line m. B is the image of B reflected in line n. The final image, B, is a translation of BC. B C m B n B Eercises In each figure, m n.find the translation image of each figure b reflecting it in line m and then in line n. 1. m n 2. B B B B m n C C B 3. m n 4. N E E T T N E T Q D Q D Q D U U m n 5. S 6. m T U U m S D n Q T U U D n U Q D 4

5 9-2 NME DTE EID Translations In each figure, a b. Determine whether figure 3 is a translation image of figure 1. Write es or no. Eplain our answer a b a b 3. a b 4. 1 a b Lesson 9-2 CDINTE GEMETY Graph each figure and its image under the given translation. 5. JKL with vertices J( 4, 4), 6. quadrilateral LMN with vertices L(4, 2), K( 2, 1), and L(2, 4) under the M(4, 1), N(0, 1), and (1, 4) under the translation (, ) ( 2, 5) translation (, ) ( 4, 3) K J K L L N L M J L N M 5

6 9-2 NME DTE EID Translations In each figure, c d. Determine whether figure 3 is a translation image of figure 1. Write es or no. Eplain our answer. 1. c d c d CDINTE GEMETY Graph each figure and its image under the given translation. 3. quadrilateral TUWX with vertices 4. pentagon DEFGH with vertices D( 1, 2), T( 1, 1), U(4, 2), W(1, 5), and X( 1, 3) E(2, 1), F(5, 2), G(4, 4), H(1, 4) under the translation under the translation (, ) ( 2, 4) (, ) ( 1, 5) X W D E F X T W U H D G E F T U H G NIMTIN Find the translation that moves the figure on the coordinate plane figure 1 figure figure 2 figure figure 3 figure 4 6

7 Name Class Date 9-2 ractice Translations Tell whether the transformation appears to be a rigid motion. Eplain reimage Image reimage Image reimage Image reimage Image In each diagram, the dashed-line figure is an image of the solid-line figure. (a) Choose an angle or point from the preimage and name its image. (b) List all pairs of corresponding sides Graph the image of each figure under the given translation. 7. T < 1, 4> (ΔBC) 8. T <3, 3> (MN) The dashed-line figure is a translation image of the solid-line figure. Write a rule to describe each translation

8 Name Class Date 9-2 ractice (continued) Translations 11. You are visiting Washington, D.C. From the merican Histor Museum ou walk 4 blocks east and 2 blocks south to the ir and Space Museum. Then ou walk 9 blocks west to the Washington Monument. Where is the Washington Monument in relation to the merican Histor Museum? 12. You and some friends go to a book fair where booths are set out in rows. You bu drinks at the refreshment stand and then walk 6 rows north and 4 rows east to the science fiction booth. Then ou walk 1 row south and 4 rows west to the children s book booth. Where is the children s book booth in relation to the refreshment stand? 13. easoning If T <15, 10> (QS) = Q S, what translation maps Q S onto QS? 14. ΔXYZ has coordinates X(3, 2), Y(4, 1), and Z(9, 8). translation maps X to X (7, 8). What are the coordinates for Y and Z for this translation? 15. Use the graph at the right. Write three different translation rules for which the image of ΔST has a verte at the origin. 16. Use the graph at the right. Write three different translation rules for which the image of ΔBCD has a verte at the origin. Graph the image of each figure under the given translation. 17. T < 3, 4> (ΔDEF) 18. T < 5, 1> (KLMN) 8

9 9-3 NME DTE EID eflections Draw eflections The transformation called a reflection is a flip of a figure in a point, a line, or a plane. The new figure is the image and the original figure is the preimage. The preimage and image are congruent, so a reflection is a congruence transformation or isometr. Eample 1 Eample 2 Construct the image of quadrilateral BCD under a reflection in line m. C B B D D m Draw a perpendicular from each verte of the quadrilateral to m. Find vertices, B,, and D that are the same distance from m on the other side of m. The image is B D. Quadrilateral DEFG has vertices D( 2, 3), E(4, 4), F(3, 2), and G( 3, 1). Find the image under reflection in the -ais. E To find an image for a D reflection in the -ais, F use the same -coordinate G and multipl the -coordinate b 1. In G smbols, (a, b) (a, b). F The new coordinates are D E D ( 2, 3), E (4, 4), F (3, 2), and G ( 3, 1). The image is D E F G. Lesson 9-1 In Eample 2, the notation (a, b) (a, b) represents a reflection in the -ais. Here are three other common reflections in the coordinate plane. in the -ais: (a, b) ( a, b) in the line : (a, b) (b, a) in the origin: (a, b) ( a, b) Eercises Draw the image of each figure under a reflection in line m M 3. m N L K m H J Q S T U m Graph each figure and its image under the given reflection. 4. DEF with D( 2, 1), E( 1, 3), 5. BCD with (1, 4), B(3, 2), C(2, 2), F(3, 1) in the -ais D( 3, 1) in the -ais D D E E F F B D C B D 9

10 9-3 NME DTE EID eflections Lines and oints of Smmetr If a figure has a line of smmetr, then it can be folded along that line so that the two halves match. If a figure has a point of smmetr, it is the midpoint of all segments between the preimage and image points. Eample Determine how man lines of smmetr a regular heagon has. Then determine whether a regular heagon has point smmetr. There are si lines of smmetr, three that are diagonals through opposite vertices and three that are perpendicular bisectors of opposite sides. The heagon has point smmetr because an line through identifies two points on the heagon that can be considered images of each other. F E B D C Eercises Determine how man lines of smmetr each figure has. Then determine whether the figure has point smmetr

11 9-3 NME DTE EID eflections Draw the image of each figure under a reflection in line CDINTE GEMETY Graph each figure and its image under the given reflection. 3. BC with vertices ( 3, 2), B(0, 1), 4. trapezoid DEFG with vertices D(0, 3), and C( 2, 3) in the origin E(1, 3), F(3, 3), and G(4, 3) in the -ais B F E E F Lesson 9-1 C B G D D G 5. parallelogram STU with vertices 6. square KLMN with vertices K( 1, 0), ( 2, 3), S(2, 4), T(2, 3) and L( 2, 3), M(1, 4), and N(2, 1) in U( 2, 4) in the line the -ais S M T S L U U T L K K M N N Determine how man lines of smmetr each figure has. Then determine whether the figure has point smmetr

12 9-3 NME DTE EID eflections Draw the image of each figure under a reflection in line CDINTE GEMETY Graph each figure and its image under the given reflection. 3. quadrilateral BCD with vertices 4. FGH with vertices F( 3, 1), G(0, 4), ( 3, 3), B(1, 4), C(4, 0), and and H(3, 1) in the line D( 3, 3) in the origin B D G H C F G H D B F 5. rectangle QST with vertices Q( 3, 2), 6. trapezoid HIJK with vertices H( 2, 5), ( 1, 4), S(2, 1), and T(0, 1) I(2, 5), J( 4, 1), and K( 4, 3) in the -ais in the -ais H I H I Q Q T T S S K J K J D SIGNS Determine how man lines of smmetr each sign has. Then determine whether the sign has point smmetr

13 Name Class Date 9-3 ractice eflections Find the coordinates of each image. 1. -ais() 2. -ais(b) 3. = 1(C) 4. = 1(D) 5. = 1(E) 6. = 2(F) Coordinate Geometr Given points M(2, 2), N(4, 1), and (3, 3), graph ΔMN and its reflection image as indicated. 7. -ais 8. -ais 9. = = 2 Cop each figure and line l. Draw each figure s reflection image across line l

14 Name Class Date 9-3 ractice (continued) eflections Cop each pair of figures. Then draw the line of reflection ou can use to map one figure onto the other Find the image of Z(1, 1) after two reflections, first across line l 1, and then across line l l 1 : = 2, l 2 : -ais 18. l 1 : = 2, l 2 : -ais 19. l 1 : = 2, l 2 : -ais 20. l 1 : = 3, l 2 : -ais 21. l 1 : = 3, l 2 : = l 1 : = 1, l 2 : = 3 Use the graph at the right for Eercises 23 and Berit lives 4 mi east of t. 147 and 1 mi north of t. 9. Jane lives 4 mi east of t. 147 and 6 mi north of t. 9. The girls want to start at Berit s house, hike to t. 147, then on to Jane s house. The want to hike the shortest distance possible. To which point on t. 147 should the walk? (Hint: First find the line of reflection if Berit s house is reflected onto Jane s house.) 24. Instead of ending the hike at Jane s house, the girls want to hike to an inn 2 mi north of Jane s house. The want to hike the shortest possible total distance, starting from Berit s house, walking to t. 147, and then to the inn. To which point on t. 147 should the walk? (Hint: First find the line of reflection if Berit s house is reflected onto the inn.) 25. oint on a coordinate grid is at (4, 3). What are the coordinates of = ()? 26. oint Z on a coordinate grid is at (3, 1). What are the coordinates of = (Z)? 27. Give an eample of a place ou ma see a geometric reflection in everda life. Eplain. 14

15 Lesson 9.4 Transformations Name eriod Date In Eercises 1 3, perform each transformation. 1. eflect TI across line. 2. otate L 270 clockwise 3. Translate ENT b about Q. the given vector. I T L T N Q E 4. BCDE and its reflected image, B D E, are shown below. Use construction tools to locate the line of reflection,. Eplain our method. D D E E C B B In Eercises 5 8, identif the tpe(s) of smmetr in each figure. 5. Equilateral triangle 6. ectangle 7. Isosceles triangle 8. Square In Eercises 9 12, draw each polgon and identif the tpe(s) of smmetr in each. Draw all lines of reflection and mark centers of rotation. 9. hombus 10. arallelogram 11. Isosceles trapezoid 12. Square 15

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17 9-4 NME DTE EID otations Draw otations transformation called a rotation turns a figure through a specified angle about a fied point called the center of rotation.to find the image of a rotation, one wa is to use a protractor. nother wa is to reflect a figure twice, in two intersecting lines. Eample 1 BC has vertices (2, 1), B(3, 4), and C(5, 1). Draw the image of BC under a rotation of 90 counterclockwise about the origin. First draw BC. Then draw a segment from, the origin, to point. Use a protractor to measure 90 counterclockwise with as one side. Draw. Use a compass to cop onto. Name the segment. epeat with segments from the origin to points B and C. B B C Eample 2 Find the image of BC under reflection in lines m and n. First reflect BC in line m. Label the image B. eflect B in line n. Label the image B. B is a rotation of BC. The center of rotation is the intersection of lines m and n. The angle of rotation is twice the measure of the acute angle formed b m and n. B m C B B n Eercises Draw the rotation image of each figure 90 in the given direction about the center point and label the coordinates. 1. Q with endpoints ( 1, 2) 2. Q with vertices ( 2, 3), Q(2, 1), and Q(1, 3) counterclockwise and (3, 2) clockwise about the point T(1, 1) about the origin Q Q Q T Q Lesson 9-3 Find the rotation image of each figure b reflecting it in line m and then in line n. 3. m n 4. n m Q C B 17

18 9-4 NME DTE EID otations otational Smmetr When the figure at the right is rotated about point b 120 or 240, the image looks like the preimage. The figure has rotational smmetr, which means it can be rotated less than 360 about a point and the preimage and image appear to be the same. The figure has rotational smmetr of order 3 because there are 3 rotations less than 360 (0, 120, 240 ) that produce an image that is the same as the original. The magnitude of the rotational smmetr for a figure is 360 degrees divided b the order. For the figure above, the rotational smmetr has magnitude 120 degrees. Eample Identif the order and magnitude of the rotational smmetr of the design at the right. The design has rotational smmetr about the center point for rotations of 0, 45, 90, 135, 180, 225, 270, and 315. There are eight rotations less than 360 degrees, so the order of its rotational smmetr is 8. The quotient is 45, so the magnitude of its rotational smmetr is 45 degrees. Eercises Identif the order and magnitude of the rotational smmetr of each figure. 1. a square 2. a regular 40-gon

19 9-4 NME DTE EID otations otate each figure about point under the given angle of rotation and the given direction. Label the vertices of the rotation image counterclockwise clockwise Q K S S J Q G H K J G H CDINTE GEMETY Draw the rotation image of each figure 90 in the given direction about the origin and label the coordinates. 3. STW with vertices S(2, 1), T(5, 1), 4. DEF with vertices D( 4, 3), E(1, 2), and W(3, 3) counterclockwise and F( 3, 3) clockwise W T W D F E D S S T Use a composition of reflections to find the rotation image with respect to lines k and m. Then find the angle of rotation for each image. F E Lesson B k 6. k C L M N B m m M L N 19

20 9-4 NME DTE EID otations otate each figure about point under the given angle of rotation and the given direction. Label the vertices of the rotation image counterclockwise clockwise N T U M S Q N U Q M T S CDINTE GEMETY Draw the rotation image of each figure 90 in the given direction about the center point and label the coordinates. 3. ST with vertices ( 3, 3), S(2, 4), 4. HJK with vertices H(3, 1), J(3, 3), and T(1, 2) clockwise about the and K( 3, 4) counterclockwise about point (1, 0) the point ( 1, 1) S T T (1, 0) S H ( 1, 1) K J H K J Use a composition of reflections to find the rotation image with respect to lines p and s. Then find the angle of rotation for each image. 5. s p 6. F E G G E F p S T T S s 7. STEMBTS paddle wheel on a steamboat is driven b a steam engine and moves from one paddle to the net to propel the boat through the water. If a paddle wheel consists of 18 evenl spaced paddles, identif the order and magnitude of its rotational smmetr. 20

21 Name Class Date 9-4 ractice otations Cop each figure and point. Draw the image of each figure for the given rotation about. Use prime notation to label the vertices of the image Cop each figure and point. Then draw the image of JK for a 180 rotation about. Use prime notation to label the vertices of the image oint is the center of regular heagon BCDEFG. Find the image of the given point or segment for the given rotation. 7. r (120, )(F) 8. r (180, )(B) 9. r (300, )( BG ) 10. r (360, )( CD ) 11. r (60, )(E) 12. r (240, )( FE ) 21

22 Name Class Date 9-4 ractice (continued) otations For Eercises 13 15, ΔBC has vertices (2, 2), B(2, 1), and C( 2, 3). 13. Graph r (90º, )( BC). 14. Graph r (180º, )( BC). 15. Graph r (270º, )( BC). 16. The vertices of QS have coordinates (6, 1), Q(4, 4), ( 4, 2), and S( 2, 3). What are the coordinates of the vertices of r (270º, )(QS)? 17. The vertices of r(90º, ) (KLMN) have coordinates K (2, 3), L (3, 2), M ( 2, 4), and N ( 4, 2). What are the coordinates of the vertices of KLMN? 18. easoning The vertices of quadrilateral BCD have coordinates (3, 4), B(4, 3), C( 3, 4), and D( 4, 3). Eplain how the transformation r (90º, )(BCD) = BCD can be used to show that the quadrilateral is square. Find the angle of rotation about D that maps the solid-line figure to the dashed-line figure pie is cut into 12 equal slices. What is the angle of rotation about the center that will map a piece of pie to a piece that is two slices awa from it? 22. ST has vertices at (3, 0), S(0, 4), and T(0, 0). What are the coordinates of the vertices of r ( 90º, T)( ST)? 23. FGH has vertices F(2, 1), G(0, 0), and H( 1, 3). What are the coordinates of the vertices of r ( 90º, G)( FGH)? 22

23 Lesson 10.1 Compositions of Transformations Name eriod Date In Eercises 1 8, name the single transformation that can replace the composition of each set of multiple transformations. 1. Translation b 4, 1, followed b 2, 3, followed b 8, 7 2. otation 60 clockwise, followed b 80 counterclockwise, followed b 25 counterclockwise all about the same center of rotation 3. eflection across vertical line m, followed b reflection across vertical line n, where n is 8 units to the right of m 4. eflection across vertical line p, followed b reflection across horizontal line q 5. eflection across vertical line n, followed b reflection across vertical line m, where n is 8 units to the right of m 6. eflection across horizontal line q, followed b reflection across vertical line p 7. Translation b 6, 0, followed b reflection across the -ais 8. eflection across the -ais, followed b translation b 6, 0 In Eercises 9 11, cop the figure onto our paper and use our geometr tools to perform the given transformation. 9. Locate, the reflected image across, and, the reflected image of across T. Find m T and give a single transformation that maps to. T 10. Locate, the reflected image across k, and, the reflected image of across. Find the distance between and k and give a single transformation that maps to. 11. Draw five glide-reflected images of the triangle. k 23

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25 Name Class Date 10-1 ractice Compositions of Transformations Find the image of each letter after the transformation m º l. 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 Graph DML and its glide reflection image. 3. ( -ais T <3, 0>)( DML) 4. ( =1 T < 1, 0>)( DML) 5. ( =2 T <0, 1>)( DML) 6. ( = T <3, 3>)( DML) 7. Lines l and m intersect at point and are perpendicular. If a point Q is reflected across l and then across m, what transformation rule describes this composition? 8. 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? 25

26 Name Class Date 10-1 ractice (continued) Compositions of Transformations Graph B and its image 'B' after a reflection first across l 1 and then across l 2. 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. 9. ( 3, 4), B( 1, 0); l 1 : = 1; l 2 : = ( 5, 2), B( 3, 6); l 1 : = 2; l 2 : = pen-ended Draw a quadrilateral on a coordinate grid. Describe a reflection, translation, rotation, and glide reflection. Then draw the image of the quadrilateral for each transformation. Identif each mapping as a translation, reflection, rotation, or glide reflection. Write the rule for each translation, reflection, rotation, or glide reflection. For glide reflections, write the rule as a composition of a translation and a reflection. 12. BC DEF 13. DEF GHF 14. DEF IJK 15. GHF IJK maps to (2, 3) b the given glide reflection. What are the coordinates of? 16. ( -ais º T <2, 0>)() 17. ( = º T <3, 3>)() 26

27 Name Class Date 10-2 ractice Congruence For each coordinate grid, identif a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image Find a congruence transformation that maps BC to DEF Determine whether the figures are congruent. If so, describe a congruence transformation that maps one to the other. If not, eplain

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