Transformations. Transformations. Reflections. Rotations. Composition of Transformations

Transcription

1 Reflections Rotations omposition of Transformations ongruence Transformations ilations Similarity Transformations Transformations Transformations transformation of a geometric figure is a mapping that results in a change in the position, shape, or size of the figure. In the game of dominoes, you often move the dominoes by sliding them, turning them or flipping them. Each of these moves is a type of transformation. translation 1

2 Transformations In a transformation, the original figure is the preimage resulting figure is the Transformations Some transformations (like the dominoes) preserve distance and angle measures. These transformations are called rigid motions To preserve distance means that the distance between any two points of the image is the same as the distance between the corresponding points of the preimage. To preserve angles means that the angles of the image have the same measures as the corresponding angles in the preimage. Transformations Which of these is a rigid motion? Transformations transformation maps every point of a figure onto its image and may be described using arrow notation ( ). Prime notation ) is sometimes used to identify image points. is the image of ilation - Size change You list the corresponding points of the preimage and image in the same order, just as you would for corresponding points in congruent figures or similar figures. 2

3 1 oes the transformation appear to be a rigid motion? Explain. Yes, it preserves the distance between consecutive points. No, it does not preserve the distance between consecutive points. 2 oes the transformation appear to be a rigid motion? Explain. Yes, distances are preserved. Yes, angle measures are preserved. oth and. No, distance are not preserved. Preimage Preimage 3 Which transformation is not a rigid motion? Reflection Translation Rotation ilation 4 Which transformation is demonstrated? Reflection Translation Rotation ilation 3

4 5 Which translation is demonstrated? Reflection Translation Rotation ilation 6 Which transformation is demonstrated? Reflection Translation Rotation ilation Translations translation the same distance in the same direction. ' = ' = ' 4

5 Translations Translations in the oordinate Plane ) = mapped to ( to describe the translation. ' ' ' ' Finding the Image of a Translation Write a translation rule that maps PQRS P'Q'R'S'. P raw ', EE' and FF'. P' S Q S' Q' R How do you know? R' 5

6 7 In the diagram, ''' is an image of. Which rule describes the translation? T<-5,-3>( ) T<5,3>( ) T<-3,-5>( ) T <3,5>( ) 8 If T<4,-6>(JKLM) = J'K'L'M', what translation maps J'K'L'M' onto JKLM? T<4,-6>(J'K'L'M') T<6,-4>(J'K'L'M') T<6,4>(J'K'L'M') T<-4,6>(J'K'L'M') 9 RSV has coordinates R(2,1), S(3,2), and V(2,6). translation maps point R to R' at (-4,8). What are the coordinates of S' for this translation? (-6,-4) (-3,2) Reflections (-3,9) (-4,13) E none of the above 6

7 is a transformation of points over a line. This line is called line of reflection. The result looks like the preimage was flipped Reflection Reflection When reflecting a figure, reflect the vertices and then draw the sides. Reflect over line r. Label the vertices of the image. lick here to see a video is not on line the perpendicular bisector of is Reflection Reflect WXYZ over line s. Label the vertices of the image. Reflection Reflect MNP over line t. Label the vertices of the image. Where is the image of N? Why? 7

8 10 Which point represents the reflection of X? point point point point E None of the above 11 Which point represents the reflection of X? point point point point E none of the above 12 Which point represents the reflection of X? point point point point E none of the above 13 Which point represents the reflection of? point point point point E none of these 8

9 14 Is a reflection a rigid motion? Yes No Reflections in the oordinate Plane Since reflections are perpendicular to and equidistant from the line of reflection, we can find the exact image of a point or a figure in the coordinate plane. Reflections in the oordinate Plane Reflections in the oordinate Plane Reflect figure the over JKLM) = J'K'L'M' 9

10 Reflections in the oordinate Plane Reflections in the oordinate Plane over the line Notation *Hint: draw line of reflection first Reflections in the oordinate Plane Find the oordinates of Each Image Reflect quadrilateral MNPQ) = M'N'P'Q' 10

11 15 The point (4,2) reflected over the x-axis has an image of. (4,2) (-4,-2) (-4,2) (4,-2) 16 The point (4,2) reflected over the y-axis has an image of. (4,2) (-4,-2) (-4,2) (4,-2) 17 has coordinates (-3,0). What would be the coordinates of ' if is reflected over the line x = 1? (-3,0) (4,0) (-3,2) (5,0) 18 The point (4,2) reflected over the line y=2 has an image of. (4,2) (4,1) (2,2) (4,-2) 11

12 line of symmetry is a line of reflection that divides a figure into 2 congruent halves. These 2 halves reflect onto each raw Lines of Symmetry Where pplicable raw Lines of Symmetry Where pplicable raw Lines of Symmetry Where pplicable 12

13 19 How many lines of symmetry does the following have? one two three none 20 How many lines of symmetry does the following have? infinitely many 21 How many lines of symmetry does the following have? none one nine infinitely many 22 How many lines of symmetry does the following have? none one two infinitely many 13

14 23 How many lines of symmetry does the following have? none one two infinitely many 24 How many lines of symmetry does the following have? none Rotations is a rigid motion that turns a figure about a point. The amount of turn is in degrees. The direction of turn is either clockwise or counterclockwise. counterclockwise clockwise about H. 14

15 Rotations rawing Rotation Images with the following properties: - The image of is itself ( is a transformation Step 1 O O O - ' ' ' O L lick here to see video rawing Rotation Images O Rotations in the oordinate Plane in the coordinate ) = r (-270, O)(x, y) = r (90, O)(x, y) L 15

16 Rotations in the oordinate Plane Rotations in the oordinate Plane in the coordinate in the coordinate ( ) r (-90, O)(x, y) = r (270, O)(x, y) Rotations in the oordinate Plane Graphing Rotation Images in the coordinate PQRS has vertices P(1, 1), Q(3, 3), R(4, 1) and S(3, 0). a) What is the graph of r (90,0)(PQRS) = P'Q'R'S'? 16

17 Graphing Rotation Images PQRS has vertices P(1, 1), Q(3, 3), R(4, 1) and S(3, 0). b) What is the graph of r (90,0)(PQRS) = P''Q''R''S''? Graphing Rotation Images PQRS has vertices P(1, 1), Q(3, 3), R(4, 1) and S(3, 0). c) What is the graph of r (270,0)(PQRS)= P'''Q'''R'''S'''? 25 Square has vertices (3,3), (-3,3), (-3, -3), and (3, -3). Which of the following images is? 26 PQRS has vertices P(1,5), Q(3, -2), R(-3, -2), and S(-5, 1). What are the coordinates of Q' after r(270,0)(q)? r (90,0)() r (180,0)() r(90,0)() r (270,0)() (-2, -3) (2,3) (-3, 2) (-3, -2) 17

18 Identifying a Rotation Image regular polygon has a center that is equidistant from its vertices. Segments that connect the center to the vertices divide the polygon into congruent triangles. You can use this fact to find rotation images of regular polygons. PENT is a regular pentagon with center O. Name the image of counterclockwise about Name the image of clockwise about O P E 27 MTH is a regular quadrilateral with center R. Name the image of M for a 180º rotation counterclockwise about R. M T H H M R M Name the image of counterclockwise about T N 28 MTH is a regular quadrilateral with center R. Name the image of for a 270º rotation clockwise about R. H R M 29 HEXGO is a regular hexagon with center M. Name the image of G for a 300º rotation counterclockwise about M. X E O H M E X M H E O G 18

19 30 HEXGO is a regular hexagon with center M. Name the image of OH for a 240º rotation clockwise about M. HE H E figure has rotational symmetry if there is at least one rotation less than or equal to 180º about a point so that the preimage is the image. G EX O M X For example, a 3-bladed fan has a rotational symmetry at 120º. X E OG G o the following have rotational symmetry? If yes, what is the degree of rotation? o the following regular shapes have rotational symmetry? If yes, what is the degree of rotation? In general, what is the rule that can be used to find the degree of rotation for a regular polygon? 19

20 31 oes the following figure have rotational symmetry? If yes, what degree? yes, 90º yes, 120º yes, 180º no 32 oes the following figure have rotational symmetry? If yes, what degree? yes, 90º yes, 120º yes, 180º no 33 oes the following figure have rotational symmetry? If yes, what degree? yes, 90º yes, 120º yes, 180º no 34 oes the following figure have rotational symmetry? If yes, what degree? yes, 18º yes, 36º yes, 72º no 20

21 omposition of Isometry When an image is used as the preimage for a second transformation it is called a composition of transformations The transformations below are isometries. is transformation that preserves distance or length. Isometry The transformations below are isometries. The composition of two or more isometries is an isometry. Glide Reflections If two figures are congruent and have opposite orientations (but are not simply reflections of each other), then there is a translation and a glide reflection composition of a glide (translation) and a reflection across a line parallel to the direction of translation. Notation for a omposition R T ( ) Note performed right to left. ' ' ' 21

22 Glide Reflections Graph the glide reflection image of Glide Reflections Graph the glide reflection image of Glide Reflections Graph the glide reflection image of omposition of Reflections Make a conjecture. by using a composition of reflections. Reflect over = -3 then over Label the first image the second 1.) What direction did XYZ slide? How is this related to the lines of reflection? 2.)How far did XYZ slide? How is this related to the lines of reflection? 22

23 35 FGHJ is translated using a composition of reflections. FGHJ is first reflected over line r then line s. How far does FGHJ slide? 36 FGHJ is translated using a composition of reflections. FGHJ is first reflected over line r then line s. Which arrow shows the direction of the slide? 5" 10" 15" 20" 37 FGHJ is translated using a composition of reflections. FGHJ is first reflected over line s then line r. How far does FGHJ slide? Rotations as omposition of Reflection 5" 10" 20" 30" Where the lines rotation is twice the acute,or right, The direction of rotation is clockwise because rotating from m to n across the acute angle is clockwise. Had the triangle reflected then m, the rotation would have been counterclockwise. 23

24 Rotations as omposition of Reflection If E is reflected over r then s: What is the angle of rotation? What is the direction of the rotation? Rotations as omposition of Reflection If E is reflected over s then r: What is the angle of rotation? What is the direction of the rotation? º º Rotations as omposition of Reflection If E is reflected over s then r: What is the angle of rotation? What is the direction of the rotation? Rotations as omposition of Reflection If E is reflected over s then r: What is the angle of rotation? What is the direction of the rotation? º º 24

25 38 If the image of is the composite of reflections over e then f, what is the angle of rotation? 40º 39 What is the direction of the rotation if the image of is the composite of reflections first over e then f? lockwise 80º 160º º ounterclockwise º 280º 40 If the image of is the composite of reflections over f then e, what is the angle of rotation? 90º 180º 41 What is the direction of rotation if the image of is the composite of reflections first over f then e? lockwise ounterclockwise 270º 360º 25

26 42 If the image of is the composite of reflections over e then f, what is the angle of rotation? 43 What is the direction of the rotation if the image of is the composite of reflections first over e then f? 40º 80º lockwise ounterclockwise 140º 160º º º ongruence ongruent Figures congruent if and only if there is a sequence of one or more rigid motions that maps one figure onto another. The composition R T ( ) = ( EF) Since compositions of rigid motions preserve angle measures and distances the corresponding sides and angles have equal measures. Fill in the blanks below: = = ontents = m = m m = m m = m 26

27 Identifying ongruence Transformations ecause compositions of rigid motions take figures to congruent figures, they are also called congruence transformations What is the congruence transformation that Z X Y Using ongruence Transformations Use congruence transformations to verify that Using ongruence Transformations is an equilateral triangle, what congruence transformation can you use that maps the triangle onto itself? 44 Which congruent transformation maps to EF? T<-5,5> 27

28 45 Which congruence transformation does not map to EF? 46 Which of the following best describe a congruence transformation that maps to EF? a reflection only a translation only a translation followed by a reflection a translation followed by a rotation 47 Quadrilateral is shown below. Which of the following transformations of E could be used to show that E is congruent to E? a reflection over a reflection over a reflection over line m a reflection over line n ilations ontents 28

29 ilation is a transformation whose pre image and image are similar. Thus, a dilation is a similarity transformation. It is, in general, a rigid motion. ilation and scale factor with the following properties: ' is a transformation ' R'=R ' ilated Pupil Every dilation has a center and a scale factor 0. The scale factor describes the size change from the original figure to the image. is itself (R' = R) - For any other point R enlargement 2 Types of ilations factor is less than one, but greater ilations The symbol for scale factor is n. dilation is an enlargement if n > 1. dilation is a reduction if 0< n < 1. What happens to a figure if n = 1? 29

30 Finding the Scale Factor ilations The dashed line figure is a dilation image of the solid-line figure. the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. The ratio of corresponding sides is image = preimage which is the scale factor ( ) of the dilation. orresponding angles are congruent. 6 Enlargement; 8 Reduction; 1/2 Reduction; 3/2 The dashed line figure is a dilation image of the solid-line figure. the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. ilations 48 Is a dilation a rigid motion? Yes No Enlargement; 2 30

31 49 Is the dilation an enlargement or a reduction? What is the scale factor of the dilation? 50 Is the dilation an enlargement or reduction? What is the scale factor of the dilation? enlargement, n = 3 enlargement, n=3 enlargement, n = 1/3 reduction, n = 3 enlargement, n = 1/3 reduction, n = 3 F' reduction, n = 1/3 F' reduction, n = 1/3 51 Is the dilation an enlargement or reduction? What is the scale factor of the dilation? 52 Is the dilation an enlargement or reduction? What is the scale factor of the dilation? enlargement, n = 2 enlargement, n = 1/2 reduction, n = 2 reduction, n = 1/2 enlargement, n = 2 enlargement, n = 3 enlargement, n = 6 not a dilation 31

32 53 The solid-line figure is a dilation of the dashed-line figure. The labeled point is the center of dilation. Find the scale factor of dilation. 54 dilation maps triangle LMN to triangle L'M'N'. MN = 14 in. and M'N' = 9.8 in. If LN = 13 in., what is L'N'? 2 3 1/2 1/3 X in. 14 in. 9.1 in. 9.8 in. rawing ilation Images ( ( EF) 0.5 in 2.5 in ' ' lick here to watch a video ' NSWER NSWER 32

33 ilations in the oordinate Plane Suppose a dilation is centered at the origin. You can find the dilation image of a point by multiplying its coordinates by the Graphing ilation Images To dilate a figure from the origin, find the dilation images of its vertices. HJK has vertices H(2, 0), J(-1, 0.5), and K(1, -2). What are the coordinates of the vertices of the image of HJK for a dilation with center (0, 0) and a scale factor 3? Graph the image and the preimage. Notation () (2, 3) ilations NOT entered at the Origin ilations ' ' dilation is NOT the origin. The Point E raw the dilation image of EF with the center of dilation at point with a scale factor 3/4. ' F lso notice 33

34 55 What is the y-coordinate of the image (8, -6) under a dilation centered at the origin and having a scale factor of 1.5? 56 What is the x-coordinate of the image (8, -6) under a dilation centered at the origin and having a scale factor of 1/2? What is the y-coordinate of the image (8, -6) under a dilation centered at the origin and having a scale factor of 1/2? 58 What is the x-coordinate of the image of (8, -6) under a dilation centered at the origin and having a scale factor of 3?

35 59 What is the x-coordinate of the image of (8, -6) under a dilation centered at the origin and having a scale factor of 1.5? 60 What is the y-coordinate of the image of (8, -6) under a dilation centered at the origin and having a scale factor of 3? What is the x-coordinate of (4, -2) under a dilation centered at (1, 3) with a scale factor of 2? 62 What is the y-coordinate of (4, -2) under a dilation centered at (1, 3) with a scale factor of 2?

36 1. hoose the correct choice to complete the sentence. Rigid motions and dilations both preserve angle measure / distance. Similarity 2. omplete the sentence by filling in the blanks. preserve distance; do not preserve distance. efine similar polygons on the lines below. ontents has vertices (-3, -3) and (-1, 1). Suppose the triangle is translated 4 units right and 2 units up and then dilated by a scale factor of 2 with the center of dilation at the origin. Sketch the resulting image of the composition of transformations. Step 1 raw the original figure. LMN has vertices L(0, 2), M(2, 2), and N(0, 1). For each similarity transformation, draw the image. 1. o R ( LMN ) Step 2 T G' (, ) H' (, ) I' (, ) Step 3 ( G'H'I') G" (, ) H"(, ) I" (, ) ( GHI) 36

37 LMN has vertices L(0, 2), M(2, 2), and N(0, 1). For each similarity transformation, draw the image. What is a composition of transformations that maps trapezoid onto trapezoid 2. o r ( LMN ) For each graph, describe the composition of transformations that maps onto FGH 1. For each graph, describe the composition of transformations that maps onto FGH H H 37

38 Similar Figures if and only if there is a similarity transformation that maps one figure onto the other. Identify the similarity transformation that maps one figure onto the other and then write a similarity statement. N N 63 Which similarity transformation maps to EF? Rotate quadrilateral counterclockwise ilate it by a scale factor of 2/3 and M coincide. the vertical angles coincide. ilate it by some scale factor ~ MNPQ 64 Which similarity transformation does not map PQR to STU? 65 Which of the following best describes a similarity transformation that maps JKP to LMP? a dilation only a rotation followed by a dilation a reflection followed by a dilation a translation followed by a dilation 38

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