Geometry. Transformations. Slide 1 / 273 Slide 2 / 273. Slide 4 / 273. Slide 3 / 273. Slide 5 / 273. Slide 6 / 273.

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1 Slide 1 / 273 Slide 2 / 273 Geometry Transformations Slide 3 / 273 Slide 4 / 273 Table of ontents Transformations Translations Reflections Rotations Identifying Symmetry with Transformations omposition of Transformations ongruence Transformations ilations Similarity Transformations PR Sample Questions click on the topic to go to that section Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: onstruct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: ttend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. dditional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Slide 5 / 273 Slide 6 / 273 Transformational Geometry Transformations were referenced in the below slide from the first unit in Geometry. This was the definition Euclid used for congruence. We just used the word "moved" instead of "transformed". Transformations Two objects are congruent if they can be moved, by any combination of translation, rotation and reflection, so that every part of each object overlaps. This is the symbol for congruence: Return to Table of ontents If a is congruent to b, this would be shown as below: a b which is read as "a is congruent to b."

2 Slide 7 / 273 Transformational Geometry Even though the idea that objects are congruent if they can be moved so that they match up is an underlying idea of geometry, it was not used directly by Euclid. Nor have we used that idea directly in this course, so far. Instead Euclid developed what we now call Synthetic Geometry, based on the idea that we can construct objects which are congruent. That's what we've been using so far. Slide 9 / 273 Transformations In 1872 Felix Klein created an approach to mathematics based on using transformations to prove objects congruent. Transformational Geometry built on the synthesis of algebra and geometry created by nalytic Geometry. It led to a further synthesis with Group Theory and other forms of mathematics which were being developed in more advanced algebra. We will explore this using transformations on the artesian plane to prove that figures are congruent or similar. Felix Klein Slide 8 / 273 Transformations The reason for that is that although two objects could look the same, that doesn't prove that they are the same. We would never accept as a proof that two figures in a drawing look like they would line up, therefore they are congruent. However, with analytic geometry, it became possible to exactly "move" an object and prove if all its points lined up with a second object. If so, the objects are congruent. This is transformational geometry. Slide 10 / 273 Transformations First, let's review the ideas of rigid transformations and then we'll develop some new notation to keep those ideas straight. Remember, rigid transformations move an object without changing its size or shape. They include translations, rotations and reflections. ilations are NOT rigid transformations since they change the size of the object, even though they preserve its shape. Slide 11 / 273 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. Slide 12 / 273 Transformations In a transformation, the original figure is the preimage, and the resulting figure is the image. In the examples below, the preimage is green and the image is pink. translation - slide rotation - turn reflection - flip

3 Slide 13 / 273 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. Slide 14 / 273 Transformations Which of these is a rigid motion? Translation- slide Rotation-turn Since distance is preserved in rigid motions, so will other measurements in the figures that rely on distance. For example, both the perimeter and area of the shape will remain the same after the completion of a rigid motion. To "preserve angles" means that the angles of the image have the same measures as the corresponding angles in the preimage. Reflection- Flip ilation - Size change Slide 15 / 273 Slide 16 / 273 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. Transformations In the diagram below, ' is the image of. ' 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. ' Δ Δ''' Δ maps onto Δ''' Note: 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. ' Preimage Image Slide 17 / oes the transformation appear to be a rigid motion? Explain. Yes, distances are preserved. Yes, angle measures are preserved. oth and. No, distances are not preserved. Slide 18 / Which transformation is not a rigid motion? Reflection Translation Rotation Preimage Image ilation

4 Slide 19 / Which transformation is demonstrated? Slide 20 / Which transformation is demonstrated? Reflection Translation Rotation ilation Reflection Translation Rotation ilation Slide 21 / Which transformation is demonstrated? Reflection Slide 22 / Quadrilateral below is reflected about line m. fter the reflection, how is the perimeter of '''' be related to the perimeter of? Translation Rotation ilation m ecause reflection is a rigid motion that preserves the distance between consecutive points, the perimeter will remain the same. ecause reflection is not a rigid motion and does not preserve the distance between consecutive points, the perimeter will change. Slide 23 / 273 Slide 24 / Quadrilateral below is rotated 180. fter the rotation, how is the area of '''' be related to the area of? Translations ecause rotation is a rigid motion that preserves the distance between consecutive points, the area will remain the same. ecause rotation is not a rigid motion and does not preserve the distance between consecutive points, the area will change. Return to Table of ontents

5 Slide 25 / 273 Translations translation is a transformation that maps all points of a figure the same distance in the same direction. Slide 26 / 273 Translations That means that any line drawn from a point on the preimage to the corresponding point on the image will be of equal length. ' ' ' = ' = ' ' ' ' ' Slide 27 / 273 Slide 28 / 273 Translations Translations That also means that the lengths of the sides of the preimage and image will be the same. nd, that the corresponding angles in the preimage and image are congruent. ' ' = '', = '', = '' ' ' m = m ' m = m ' m = m ' ' ' Slide 29 / 273 Translations So far, so good...but if I'm given these two triangles, how do I prove them congruent based on using transformations. That's made possible by adding in nalytic Geometry through the use of a artesian plane. ' Slide 30 / 273 Translations If I can show that translating each of the vertices of Δ in the same way results in them lining up with all the vertices of Δ''' then we have proven those Δs. ' ' ' ' '

6 Slide 31 / 273 Translations In this case, we show below that each preimage vertex is translated to each image vertex by translating up 4 units (positive y-direction) and 10 units right (positive x-direction). Slide 32 / 273 Translations While, strictly speaking, we need to do that for every point, not just every vertex, we can be confident that once the endpoints line up, so will all of the points on the line segements that connect them. ' ' ' ' ' ' Slide 33 / 273 Translations The notation to describe this translation is: T <10,4>: (Δ) Δ''' Slide 34 / 273 Translations in the oordinate Plane is translated 9 units right and 4 units down. The letter "T" means "translation" and the two numbers in its subscript are the units moved along the x-axis and the the y- axis in order to transform Δ into Δ'''. The symbol indicates that transformation. ' ' ' Each (x, y) pair in is mapped to (x + 9, y - 4). You can use the function notation T <9, -4>: () = '''' to describe the translation. ' ' ' ' The order of the subscripts is the same as it is for an ordered pair (x,y). Slide 35 / 273 Finding the Image of a Translation What are the coordinates of the preimage and image vertices? ( ) ' ( ) E ( ) E' ( ) F ( ) F' ( ) Graph the image of ΔEF. raw ', EE' and FF'. What relationships exist among these three segments? How do you know? T <-2, 5> : (ΔEF) Δ'E'F' E F Slide 36 / 273 Writing a Translation Rule Write a translation rule that maps PQRS P'Q'R'S'. P S P' Q S' Q' R R'

7 Slide 37 / In the diagram, Δ''' is an image of Δ. Which rule describes the translation? T <-5,-3> : (Δ) T <5,3> : (Δ) T <-3,-5> : (Δ) T <3,5> : (Δ) Slide 38 / 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') Slide 39 / 273 Slide 40 / ΔRSV 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) (-3, 9) (-4, 13) Reflections Reflections ctivity Lab (lick for link to lab) E none of the above Return to Table of ontents Slide 41 / 273 Reflection reflection is a transformation of points over a line. This line is called the line of reflection. In this case, the line of reflection is line m. The result looks like the preimage was flipped over the line; the preimage and the image have opposite orientations. m ' m Slide 42 / 273 Reflection ny point of the image which lies on the line of reflection maps onto itself. In this case, = '. ' line of reflection ' ' ' '

8 Slide 43 / 273 Reflection Lines which connect corresponding points on the image and preimage are bisected by, and are perpendicular to, the line of reflection. In this case, line m is the perpendicular bisector of ' and '. Slide 44 / 273 Reflection The reflection across m that maps Δ Δ''' is written as R m : Δ Δ'' The subscript after "R" is the name of the line of reflection. ' ' m ' ' m ' ' Slide 45 / 273 When reflecting a figure, reflect the vertices and then draw the sides. Reflect over line r. Label the vertices of the image. Reflection r lick here to see a video Slide 46 / 273 Reflection Reflect WXYZ over line s. Label the vertices of the image. W Z X Y s Hint: Turn page so line of symmetry is vertical Slide 47 / 273 Reflection Slide 48 / Which point represents the reflection of X? Reflect MNP over line t. Label the vertices of the image. M Where is the image of N? Why? point point point point X t N P E None of the above

9 Slide 49 / Which point represents the reflection of X? Slide 50 / Which point represents the reflection of X? point point point point point E none of the above X point point point E none of the above X Slide 51 / Which point represents the reflection of? Slide 52 / Is a reflection a rigid motion? point Yes point point point No E none of these Slide 53 / 273 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. Slide 54 / 273 Reflections in the oordinate Plane Reflect,, & over the y-axis. How do the coordinates of each point change when the point is reflected over the y-axis? R y-axis :() ' R y-axis :() ' R y-axis :() '

10 Slide 55 / 273 Reflections in the oordinate Plane Slide 56 / 273 Reflections in the oordinate Plane R x-axis :(JKLM) (J'K'L'M') M R y-axis :(JKLM) (J'K'L'M') M raw the vertices and then the image. K L raw the vertices and then the image. K L How do the coordinates of a point change when it is reflected over the x-axis? J How do the coordinates of a point change when it is reflected over the y-axis? J Slide 57 / 273 Reflections in the oordinate Plane R y=x :(,,, ) (', ', ', ') raw the line y = x. Then draw the points ', ', ', and ' How do the coordinates of a point change when it is reflected over the line y = x? Slide 58 / 273 Reflections in the oordinate Plane R x=2 :(Δ) :(Δ''') raw Δ''' an you determine how the coordinates of a point change when it is reflected over a line x = a? *Hint: draw line of reflection first Slide 59 / 273 Reflections in the oordinate Plane Slide 60 / 273 Find the oordinates of Each Image R y=-3 :(MNPQ) (M'N'P'Q') raw M'N'P'Q' an you determine how the coordinates of a point change when it is reflected over a line y = b? M Q N P 1. R x-axis() 2. R y-axis() 3. R y=1() 4. R x=-1() 5. R y=x(e) F E 6. R x=-2(f)

11 Slide 61 / The point (4, 2) reflected over the x-axis has an image of. (4, 2) (-4, -2) (-4, 2) (4, -2) Slide 62 / The point (4, 2) reflected over the y-axis has an image of. (4, 2) (-4, -2) (-4, 2) (4, -2) Slide 63 / 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) Slide 64 / The point (4, 2) reflected over the line y = 2 has an image of. (4, 2) (4, 1) (2, 2) (4, -2) Slide 65 / Triangle is graphed in the coordinate plane with vertices (1, 1), (3, 4), and (-1, 8) as shown in the figure. Part Triangle will be reflected across the line y = 1 to form Δ'''. Select all quadrants of the xy-coordinate plane that will contain at least one vertex of Δ'''. Slide 66 / Triangle is graphed in the coordinate plane with vertices (1, 1), (3, 4), and (-1, 8) as shown in the figure. Students type their answers here Part Triangle will be reflected across the line y = 1 to form Δ'''. What are the coordinates of '? PR Released Question - EOY - alculator Section - Question #14 PR Released Question - EOY - alculator Section - Question #14

12 Slide 67 / 273 Slide 68 / 273 Rotations Rotations ctivity Lab (lick for link to lab) Return to Table of ontents Rotations are the third, and final, rigid transformation. rotation is usually described by the number of degrees (including direction) and the center of rotation. In this case, we will rotate xº clockwise around the point P. P Rotations P x ' ' ' Slide 69 / 273 Rotations Slide 70 / 273 Rotations The center of rotation, P, is never changed by the rotation: P = P'. This is always true, whether P is part of the figure or outside of it. This can be useful in solving problems. P x ' ' ' For any point other than P: P' P. That means that the distance from the center of rotation to a point's preimage is the same as the distance to that point's image. P x ' ' ' Slide 71 / 273 Rotations Slide 72 / 273 Rotations The measure of the angle whose vertices are a point's preimage, the center of rotation and the point's image is equal to the measure of the angle of rotation. The m P' = xº P x ' ' Rotations in the counterclockwise direction are considered positive. Rotations in the clockwise direction are considered negative. P x ' ' ' '

13 Slide 73 / 273 Slide 74 / 273 Rotations rawing Rotation Images counterclockwise rotation of xº about point P is written: r (x,p) In this case, since the rotation is clockwise, the transformation is: r (-x,p) :(Δ) (Δ''') ounterclockwise rotations are considered positive. So, mapping this image to its preimage would be written as: P x ' ' ' O L r (70,) :(ΔLO) (ΔL'O'') r (x, P):(Δ''') (Δ) Slide 75 / 273 Slide 76 / 273 rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') O Step 1 raw a line from to any vertex, for instance. O L L Step 1 raw a line from to any vertex, for instance. Slide 77 / 273 Slide 78 / 273 rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') O O L L Step 2 Place a protractor along the line with it's center at the center of rotation,, and draw a ray at a 70º angle from line O (W since this rotation is positive). Step 3. Use a compass to make a line segment on the new ray the same length as. ll angle measures must be centered on, the center of rotation.

14 Slide 79 / 273 rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') Slide 80 / 273 rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') O ' O ' L L L' Step 4. Label the new point '. Step 4. Repeat for all vertices. Slide 81 / 273 rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') Slide 82 / 273 rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') O ' O ' L O' L' L O' L' Step 4. Repeat for all vertices. Step 5. Now, connect the vertices of the rotated object. Slide 83 / 273 rawing Rotation Images r (70,) :(ΔLO) (ΔL'O'') Slide 84 / 273 rawing Rotation Images r (100,) :(ΔLO) (ΔL'O'') O ' O L O' Step 5. Note that you can rotate the objects so that they overlap perfectly...or pretty perfectly. If you would like to see the steps of constructing a rotation through a video, click the link to the right. L' lick here to see video L

15 Slide 85 / 273 rawing Rotation Images raw the image created by r (-80,) :(ΔLO). Slide 86 / 273 Rotations in the oordinate Plane r (90,O): (x, y) (-y, x) ' (-4, 3) (3, 4) O L Slide 87 / 273 Rotations in the oordinate Plane 90º rotation in one direction around the origin has the same effect as a 270º rotation in the opposite direction. Slide 88 / 273 Rotations in the oordinate Plane r (180,O): (x, y) (-x, -y) r (90, O): (x, y) = r (-270, O): (x, y) So, it's also true that... ' (-4, 3) (3, 4) (3, 4) r (-270, O): (x, y) = (-y, x) ' (-3, -4) Slide 89 / 273 Rotations in the oordinate Plane nd since a 180º rotation around the origin has the same effect as a -180º rotation, then Slide 90 / 273 Rotations in the oordinate Plane r (180,O): (x, y) = r -180,O: (x, y) r (-90,O): (x, y) (y, -x) So, therefore, r (-180,O): (x, y) (-x, -y) (3, 4) (3, 4) ' (-3, -4) ' (4, -3)

16 Slide 91 / 273 Rotations in the oordinate Plane 90º rotation in one direction around the origin is equal to a 270º rotation in the opposite direction. r (270,O): (x, y) = r (-90,O): (x, y) Slide 92 / 273 Rotations in the oordinate Plane r (-90,O): (x, y) (y, -x) So, the same rule applies r (270, O): (x, y) (y, -x) (3, 4) 90º rotation in one direction around the origin is equal to a 270º rotation in the opposite direction. (3, 4) r (270,O): (x, y) = r (-90,O): (x, y) ' (4, -3) So, the same rule applies ' (4, -3) r (270, O): (x, y) (y, -x) Slide 93 / 273 Rotations in the oordinate Plane Slide 94 / 273 Graphing Rotation Images PQRS has vertices P(1, 1), Q(2, 6), R(5,8) and S(8, 4). r (360,O): (x, y) = (x, y) a) raw PQRS ' (3, 4) Slide 95 / 273 Graphing Rotation Images PQRS has vertices P(1, 1), Q(2, 7), R(5,9) and S(9, 4). Slide 96 / 273 Graphing Rotation Images PQRS has vertices P(1, 1), Q(2, 7), R(5,9) and S(9, 4). r (90, O) :(PQRS) (P'Q'R'S') b) raw P'Q'R'S' Q P R S r (90, O) :(PQRS) (P'Q'R'S') b) raw P'Q'R'S' r (90, O) :(x, y) (-y, x) P(1,1) P'(-1,1) Q(2,7) Q'(-7,2) R(5,9) R'(-9,5) S(9,4) S'(-4,9) R' Q' S' P' Q P R S

17 Slide 97 / Square has vertices (3, 3), (-3, 3), (-3, -3), and (3, -3). Which of the following images has the same location as? r (90, O) :() Slide 98 / 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, O) :(Q)? (-2, -3) r (180, O) :() r (-90, O) :() r (270, O) :() (2, 3) (-3, 2) (-3, -2) Slide 99 / 273 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. a) Name the image of E for a 72 rotation counterclockwise about O. b) Name the image of P for a 216 rotation clockwise about O. O P E Slide 100 / MTH is a regular quadrilateral with center R. Name the image of M for a 180º rotation counterclockwise about R. M T H H T R M c) Name the image of P for a 144 rotation counterclockwise about O. T N Slide 101 / MTH is a regular quadrilateral with center R. Name the image of T for a 270º rotation clockwise about R. M H M Slide 102 / HEXGO is a regular hexagon with center M. Name the image of G for a 300º rotation counterclockwise about M. H E T TH R X E O M X HM T H E O G

18 Slide 103 / HEXGO is a regular hexagon with center M. Name the image of OH for a 240º rotation clockwise about M. HE G EX X E OG O H G M E X 29 Triangle is shown in the xy-coordinate plane. Will remain the same Slide 104 / 273 The triangle is rotated 180 counterclockwise around the point (3, 4) to create triangle '''. Indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle. The coordinates of ' Will not remain the same PR Released Question - EOY - alculator Section - Question #4 - SMRT Response Format Slide 105 / 273 Slide 106 / Triangle is shown in the xy-coordinate plane. The triangle is rotated 180 counterclockwise around the point (3, 4) to create triangle '''. Indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle. The coordinates of ' Will remain the same Will not remain the same 31 Triangle is shown in the xy-coordinate plane. The triangle is rotated 180 counterclockwise around the point (3, 4) to create triangle '''. Indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle. The perimeter of ''' Will remain the same Will not remain the same PR Released Question - EOY - alculator Section - Question #4 - SMRT Response Format Slide 107 / 273 PR Released Question - EOY - alculator Section - Question #4 - SMRT Response Format Slide 108 / Triangle is shown in the xy-coordinate plane. The triangle is rotated 180 counterclockwise around the point (3, 4) to create triangle '''. Indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle. The area of ''' Will remain the same Will not remain the same 33 Triangle is shown in the xy-coordinate plane. The triangle is rotated 180 counterclockwise around the point (3, 4) to create triangle '''. Indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle. The measure of ' Will remain the same Will not remain the same PR Released Question - EOY - alculator Section - Question #4 - SMRT Response Format PR Released Question - EOY - alculator Section - Question #4 - SMRT Response Format

19 Slide 109 / 273 Slide 110 / Triangle is shown in the xy-coordinate plane. The triangle is rotated 180 counterclockwise around the point (3, 4) to create triangle '''. Indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle. Identifying Symmetry with Transformations The slope of '' Will remain the same Will not remain the same PR Released Question - EOY - alculator Section - Question #4 - SMRT Response Format Return to Table of ontents Slide 111 / 273 Symmetry with Transformations Shapes that can be mapped onto itself with one or more transformations are said to have symmetry. Slide 112 / 273 Line of Symmetry line of symmetry is a line of reflection that divides a figure into 2 congruent halves. These 2 halves reflect onto each other. The two main types of symmetry are line symmetry (or reflection symmetry) and rotational symmetry. Let's examine each of these terms in more detail. Slide 113 / 273 raw Lines of Symmetry Where pplicable E F Slide 114 / 273 raw Lines of Symmetry Where pplicable M N O P Q R

20 Slide 115 / 273 raw Lines of Symmetry Where pplicable Slide 116 / How many lines of symmetry does the following have? one two three none Slide 117 / How many lines of symmetry does the following have? Slide 118 / How many lines of symmetry does the following have? infinitely many none one nine infinitely many Slide 119 / How many lines of symmetry does the following have? Slide 120 / How many lines of symmetry does the following have? none none one two J one two H infinitely many infinitely many

21 Slide 121 / How many lines of symmetry does the following have? none Slide 122 / 273 Rotational Symmetry figure has rotational symmetry if there is at least one rotation less than 360º about a point so that the preimage is the image. For example, a 3-bladed fan has rotational symmetry. There are 2 copies of the 3-bladed fan to the right. Separate them and rotate one of them to see the rotational symmetry. It occurs at 120º & 240. circle has infinite rotational symmetry. Slide 123 / 273 Rotational Symmetry o the following have rotational symmetry? If yes, what are the degrees of rotation? List all angles that apply. Slide 124 / 273 Rotational Symmetry o the following regular shapes have rotational symmetry? If yes, what are the degrees of rotation? List all angles that apply. a. c. b. In general, what is the rule that can be used to find the smallest degree of rotation for a regular polygon? Slide 125 / oes the following figure have rotational symmetry? If yes, what is the smallest degree of rotational symmetry? Slide 126 / oes the following figure have rotational symmetry? If yes, what is the smallest degree of rotational symmetry? yes, 90º yes, 120º yes, 180º no S yes, 90º yes, 120º yes, 180º no

22 Slide 127 / oes the following figure have rotational symmetry? If yes, what is the smallest degree of rotational symmetry? yes, 90º yes, 120º yes, 180º no Slide 128 / oes the following figure have rotational symmetry? If yes, what is the smallest degree of rotational symmetry? yes, 18º yes, 36º yes, 72º no Slide 129 / The figure shows two perpendicular lines s and r intersecting at point P in the interior of a trapezoid. Line s bisects both bases of the trapezoid. Which transformation will always carry the figure onto itself? Select all that apply. s a reflection across line r a reflection across line s P r Slide 130 / 273 Symmetry Extension with onstructions Symmetries can be used to construct regular polygons, because of their unique shape. The example below shows how 1 isosceles triangle can be rotated (or reflected) repeatedly to create the regular octagon. a rotation 90 clockwise about point P a rotation 180 clockwise about point P E a rotation 270 clockwise about point P Using these principles, let's complete 3 constructions. PR Released Question - EOY - Non-alculator - Question #2 Slide 131 / 273 onstruction of an Equilateral Triangle n equilateral triangle is the foundation for some other constructions that we will be encountering later on, so we're going to start with constructing this shape. Slide 132 / 273 onstruction of an Equilateral Triangle 2. Take your compass and place the tip on one endpoint & make an arc. Your arc should be about 90 (1/4 of a circle). 1. onstruct a segment of any length.

23 Slide 133 / 273 onstruction of an Equilateral Triangle 3. Take your compass and place the tip on the other endpoint & make an arc that intersects your 1st arc. Slide 134 / 273 onstruction of an Equilateral Triangle 4. Make a point where your two arcs intersect. onnect your 3 points. Slide 135 / 273 onstruction of an Equilateral Triangle Try this! reate an equilateral triangle using the segment below. 1) Slide 136 / 273 onstruction of an Equilateral Triangle Try this! reate an equilateral triangle using the segment below. 2) Slide 137 / 273 onstruction of an Equilateral Triangle w/ rod, string & pencil 1. onstruct a segment of any length. Slide 138 / 273 onstruction of an Equilateral Triangle w/ rod, string & pencil 2. Place your rod on one endpoint & the pencil on the other. Extend your pencil to make an arc. Your arc should be about 90 (1/4 of a circle).

24 Slide 139 / 273 onstruction of an Equilateral Triangle w/ rod, string & pencil Slide 140 / 273 onstruction of an Equilateral Triangle w/ rod, string & pencil 3. Switch the places of your rod & pencil tip. Make an arc that intersects your 1st arc. 4. Make a point where your two arcs intersect. onnect your 3 points. Slide 141 / 273 onstruction of an Equilateral Triangle w/ rod, string & pencil Try this! reate an equilateral triangle using the segment below. 3) Slide 142 / 273 onstruction of an Equilateral Triangle w/ rod, string & pencil Try this! reate an equilateral triangle using the segment below. 4) Slide 143 / 273 Slide 144 / 273 onstruction of a Regular Hexagon Video emonstrating onstructing Equilateral Triangles using ynamic Geometric Software 1. onstruct a segment of any length. lick here to see video

25 Slide 145 / 273 onstruction of a Regular Hexagon 2. reate a circle with your center at one endpoint. It doesn't matter which one you use. Slide 146 / 273 onstruction of a Regular Hexagon 3. Take your compass and place the tip on the other endpoint & make an arc that intersects your 1st arc. This point of intersection creates an equilateral triangle. Slide 147 / 273 onstruction of a Regular Hexagon 4. Repeat step #3 using the new point of intersection as the vertex. This point of intersection creates another equilateral triangle. Slide 148 / 273 onstruction of a Regular Hexagon 5. onnect all of the intersection points created by the compass arcs and the circle to create your regular hexagon (shown in red). Keep repeating this step until you reach point. Note: The additional diagonals are drawn (in purple) to show how a regular hexagon is created by rotating an equilateral triangle 60 degrees repeatedly (or as a result of continuous reflections with the radii of the circle). Slide 149 / 273 onstruction of a Regular Hexagon Try this! onstruct a regular hexagon using the segment below. 5) Slide 150 / 273 onstruction of a Regular Hexagon Try this! onstruct a regular hexagon using the segment below. 6)

26 Slide 151 / 273 Slide 152 / 273 Videos emonstrating onstructing a Regular Hexagon using ynamic Geometric Software lick here to see hexagon video omposition of Transformations Return to Table of ontents Slide 153 / 273 omposition of Transformations When an image is used as the preimage for a second transformation it is called a composition of transformations. The example below shows a reflection about the y-axis followed by a rotation of 90 clockwise about the origin. J M K L L'' M'' L' K' J'' J' K'' Slide 155 / 273 Glide Reflections Graph the glide reflection image of 1.) R x-axis T <-2, 0> : (Δ) M' Δ. Slide 154 / 273 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 reflection that will map one onto the other. glide reflection is the composition of a glide (translation) and a reflection across a line parallel to the direction of translation. Notation for a omposition R y = -2 T <1, 0> (Δ) Note: The " " above means the word "after". So, the notation above can be read as "a reflection about the line y = -2 after a translation 1 unit to the right". Therefore, transformations are performed right to left. ' ' Slide 156 / 273 Glide Reflections Graph the glide reflection image of 2.) R y-axis T <0, -3> : (Δ) ' Δ.

27 Slide 157 / 273 Glide Reflections Graph the glide reflection image of 3.) R x = -1 T <-2, -3> : (Δ) Δ. X Y Z Slide 158 / 273 omposition of Reflections Translate ΔXYZ by using a composition of reflections. Reflect over x = -3 then over x = 4. Label the first image ΔX'Y'Z' and the second ΔX"Y"Z". 1.) What direction did ΔXYZ slide? How is this related to the lines of reflection? Make a conjecture. 2.) How far did ΔXYZ slide? How is this related to the lines of reflection? Slide 159 / FGHJ is translated using a composition of reflections. FGHJ is first reflected over line r then line s. How far does FGHJ slide? r 5" 10" 15" 20" F G H J 10" s Slide 160 / 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? r F G H J 10" s Slide 161 / FGHJ is translated using a composition of reflections. FGHJ is first reflected over line s then line r. How far does FGHJ slide? r 5" 10" 20" 30" F G H J 10" s Slide 162 / FGHJ is translated using a composition of reflections. FGHJ is first reflected over line s then line r. Which arrow shows the direction of the slide? r F G H J 10" s

28 Slide 163 / 273 Rotations as omposition of Reflections Where the lines of reflection intersect, P, is center of rotation. P 80º m ' 160º ' " ' " n " The amount of rotation is twice the acute, or right, angle formed by the lines of reflection. Slide 164 / 273 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? E 20º r s The direction of rotation is clockwise because rotating from m to n across the acute angle is clockwise. Had the triangle reflected over n then m, the rotation would have been counterclockwise. Slide 165 / 273 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? E 20º r s Slide 166 / 273 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? 90º E r s Slide 167 / 273 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? s 110º E r Slide 168 / If the image of Δ is the composite of reflections over e then f, what is the angle of rotation? 40º 80º 160º 280º 40º e f

29 Slide 169 / What is the direction of the rotation if the image of Δ is the composite of reflections first over e then f? Slide 170 / If the image of Δ is the composite of reflections over f then e, what is the angle of rotation? lockwise ounterclockwise 40º e f 90º 180º 270º e 360º f Slide 171 / What is the direction of rotation if the image of Δ is the composite of reflections first over f then e? lockwise ounterclockwise e Slide 172 / If the image of Δ is the composite of reflections over e then f, what is the angle of rotation? 280º 140º 80º 40º 140º e f f Slide 173 / What is the direction of the rotation if the image of Δ is the composite of reflections first over e then f? lockwise ounterclockwise 140º e Slide 174 / Triangle has vertices at (1, 2), (4, 6), and (4, 2) in the coordinate plane. The triangle will be reflected over the x-axis and then rotated 180 about the origin to form Δ'''. What are the vertices of Δ'''? '(1, -2), '(4, -6), '(4, -2) '(-1, -2), '(-4, -6), '(-4, -2) '(-1, 2), '(-4, 6), '(-4, 2) '(1, 2), '(4, 6), '(4, 2) PR Released Question - EOY - alculator - Question #12

30 Slide 175 / Quadrilateral and EFGH are shown in the coordinate plane. Part : Quadrilateral EFGH is the image of after a transformation or sequence of transformations. Which could be the transformation or sequence of transformations? Select all that apply? translation 3 units to the right followed by a reflection across the x-axis rotation of 180 about the origin a translation of 12 units downward, followed by a reflection across the y-axis a reflection across the y-axis, followed by a reflection across the x-axis E a reflection across the line with equation y = x. PR Released Question - EOY - alculator - Question #21 Slide 177 / 273 Slide 176 / Quadrilateral and EFGH are shown in the coordinate plane. Part : Quadrilateral will be reflected across the x-axis and then rotated 90 clockwise about the origin to create quadrilateral ''''. What will be the y-coordinate of '? PR Released Question - EOY - alculator - Question #21 Slide 178 / 273 Isometry n isometry is a transformation that preserves distance or length. The transformations below are isometries. ongruence Transformations Translation Rotation Reflection Return to Table of ontents Slide 179 / 273 Isometry If two or more isometries are used in a composition, the result will also be an isometry. Glide Reflection Slide 180 / 273 ongruent Figures Two figures are congruent if and only if there is a sequence of one or more rigid motions that maps one figure onto another. The composition R m T <2, 3> :(Δ) = (ΔEF) Since compositions of rigid motions preserve angle measures and distances the corresponding sides and angles have equal measures. Fill in the blanks below: = ' = = ' ' E F m = m m = m m m = m

31 Slide 181 / 273 Identifying ongruence Transformations ecause compositions of rigid motions take figures to congruent figures, they are also called congruence transformations, which is another name for "isometries". Slide 182 / 273 Using ongruence Transformations Use congruence transformations to verify that Δ ΔEF What is the congruence transformation, or isometry, that maps ΔXYZ to Δ? X Z Y s Slide 183 / 273 Using ongruence Transformations To show that Δ is an equilateral triangle, what congruence transformation can you use that maps the triangle onto itself? Explain. Slide 184 / Which congruent transformation maps Δ to ΔFE? T <-5,5> r (180, O) R x-axis T <5, 0> n P p R x-axis r (90, O) m Slide 185 / Which congruence transformation does not map Δ to ΔEF? r 180, O T <0, 6> R y-axis Slide 186 / Which of the following best describe a congruence transformation that maps Δ to ΔEF? a reflection only a translation only a translation followed by a reflection a translation followed by a rotation F E

32 Slide 187 / 273 Slide 188 / Quadrilateral is shown below. Which of the following transformations of ΔE could be used to show that ΔE is congruent to ΔE? a reflection over a reflection over line m a reflection over a reflection over line n m ilations E n Return to Table of ontents Slide 189 / 273 ilation dilation is a transformation whose preimage and image are similar. Thus, a dilation is a similarity transformation. It is not, in general, a rigid motion. Slide 190 / 273 ilation dilation with center R and scale factor n, n > 0, is a transformation with the following properties: ' ' R'=R ' Pupil ilated Pupil Every dilation has a center and a scale factor n, n > 0. The scale factor describes the size change from the original figure to the image. - The image of R is itself (R' = R) - For any other point, ' is on R - R' = n R or n = R' R - Δ ~ Δ''' dilation is an enlargement if the scale factor is greater than 1. Slide 191 / Types of ilations dilation is a reduction if the scale factor is less than one, but greater than 0. Slide 192 / 273 ilations The symbol for scale factor is n. dilation is an enlargement if n > 1. dilation is a reduction if 0 < n < 1. scale factor of 2 scale factor of 1.5 scale factor of 3 scale factor of 3/4 scale factor of 1/2 scale factor of 1/4 What happens to a figure if n = 1?

33 Slide 193 / 273 Slide 194 / Finding the Scale Factor ' 18 ' ilations The dashed line figure is a dilation image of the solid-line figure. is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. 3 ' The ratio of corresponding sides is image = preimage which is the scale factor (n) of the dilation. orresponding angles are congruent. 9 ' 1 2 Reduction; NSWER1/2 Reduction; NSWER2/3 6 3 Enlargement; 3; length NSWER of entire segment = Slide 195 / 273 Slide 196 / 273 The dashed line figure is a dilation image of the solid-line figure. is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. 4 4 ilations 63 Is a dilation a rigid motion? Yes No 8 4 Enlargement; NSWER 2 Enlargement; NSWER 2 Slide 197 / Is the dilation an enlargement or a reduction? What is the scale factor of the dilation? Slide 198 / 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 8 F enlargement, n = 1/3 reduction, n = 3 F' 8 reduction, n = 1/3 24 F' reduction, n = 1/3 F 24

34 Slide 199 / Is the dilation an enlargement or reduction? What is the scale factor of the dilation? Slide 200 / 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 H H' 2 4 enlargement, n = 2 enlargement, n = 3 enlargement, n = 6 not a dilation R 5 R' Slide 201 / 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. Slide 202 / 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. Slide 203 / 273 Slide 204 / 273 Steps rawing ilation Images raw the dilation image Δ (2, X)(Δ) 1. Use a straightedge to construct ray X. 2. Use a compass to measure X. 1 in 3.onstruct X' by constructing a congruent segment on ray X so that X' is twice the distance of X. 4.Repeat steps 1-3 with points and. 0.5 in X 2.5 in 5 in ' ' lick here to watch a video ' a. ) (0.5, )() b. ) (2, )(ΔEF) ' raw Each ilation Image ' ' NSWER ' Note: When dilating by a scale factor of 0.5, you need to find the midpoint of your segments. E E E' ' NSWER F F F'

35 Slide 205 / 273 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 scale factor. Scale factor 2, (x, y) (2x, 2y) Notation 2 () = ' ' (4, 6) (2, 3) Slide 206 / 273 Graphing ilation Images To dilate a figure from the origin, find the dilation images of its vertices. ΔHJK has vertices H(2, 0), J(-1, 0.5), and K(1, -2). What are the coordinates of the vertices of the image of ΔHJK for a dilation with center (0, 0) and a scale factor 3? Graph the image and the preimage. Slide 207 / 273 ilations NOT entered at the Origin Slide 208 / 273 ilations ' ' ' ' In this example, the center of dilation is NOT the origin. The center of dilation (-2, -2) is a vertex of the original figure. This is a reduction with scale factor 1/2. Point and its image are the same. It is important to look at the distance from the center of dilation, to the other points of the figure. If = 6, then '' = 6/2 = 3. lso notice = 4 and '' = 2, etc. F E raw the dilation image of ΔEF with the center of dilation at point with a scale factor 3/4. Slide 209 / 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? Slide 210 / 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?

36 Slide 211 / 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? Slide 212 / 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? Slide 213 / 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? Slide 214 / 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? Slide 215 / What is the x-coordinate of (4, -2) under a dilation centered at (1, 3) with a scale factor of 2? Slide 216 / What is the y-coordinate of (4, -2) under a dilation centered at (1, 3) with a scale factor of 2? raw a graph w/ your two points. Hint: raw a graph w/ your two points. Hint:

37 Slide 217 / In the coordinate plane, Δ has vertices at (1, -2), (1, 0.5), (2, 1); and ΔEF has vertices at (4, -3), E(4, 2), F(6, 3). Select an answer from each group of choices to correctly complete the sentence. The triangles are similar because ΔEF is the image of Δ under a dilation with center (0, 0) (1, -2) (-2, -1) and scale factor 2 E 3 F 4 PR Released Question - EOY - Non-alculator Section - Question #7 Slide 219 / 273 Slide 218 / The figure shows and PQ intersecting at point. '' and P'Q' will be the images of lines and PQ, respectively, under a dilation with center P and scale factor 2. parallel to perpendicular to the same line as and line P'Q' will be PQ. parallel to E perpendicular to F the same line as P Select one of the choices from each group given below to complete the sentence. Line '' will be Q PR Released Question - EOY - alculator Section - Question #6 Slide 220 / In the coordinate plane shown, Δ has vertices (-4, 6), (2, 6) and (2, 2) 81 In the Students coordinate type their plane answers shown, here Δ has vertices (-4, 6), (2, 6) and (2, 2) What is the scale factor that will carry Δ to ΔEF? PR Released Question - EOY - alculator Section - Question #9 Slide 221 / 273 What is the center of dilation that will carry Δ to ΔEF? PR Released Question - EOY - alculator Section - Question #9 Slide 222 / In the coordinate plane, line p has a slope of 8 and a y-intercept of (0, 5). Line r is the result of dilating line p by a scale factor of 3 with a center (0, 3). What is the slope and y-intercept of line r? Line r has a slope of 5 and y-intercept of (0, 2). Line r has a slope of 8 and y-intercept of (0, 5). Line r has a slope of 8 and y-intercept of (0, 9). Line r has a slope of 11 and y-intercept of (0, 8). 83 Line segment with endpoints (4, 16) and (20, 4) lies in the coordinate plane. The segment will be dilated with a scale factor of 3/4 and a center at the origin to create ''. What will be the length of ''? PR Released Question - P - Non-alculator Section - Question #3 PR Released Question - P - alculator Section - Question #4

38 Slide 223 / 273 Released PR Exam Question Slide 224 / 273 This is a great problem and draws on a lot of what we've learned. Try it in your groups. Then we'll work on it step by step together by asking questions that break the problem into pieces. The following question from the released PR exam uses what we just learned and combines it with what we learned earlier to create a challenging question. Please try it on your own. Then we'll go through the process we used to solve it. PR Released Question - P - alculator Section - Question #9 Slide 225 / What have we learned that will help solve this problem? onstruction of a ilation w/ a compass & straightedge The efinition of a ilation Similar triangles have proportional corresponding parts ll of the above Slide 226 / What can you see from this drawing? Δ and Δ'' are similar The value of the scale factor, k The corresponding sides of Δ and Δ'' are proportional and only ' ' Slide 227 / What should be the first statement in our proof? '' = k '' is the image of after a dilation centered at point with a scale factor of k, where k > 0 The image of point is itself. That is, ' =. For any point P other than, the point P' is on P, and P' = k P. Slide 228 / Since we know that a dilation is occurring, what should be the second statement in our proof? The image of point is itself. That is, ' =. ' = k ' = k and only

39 Slide 229 / What reason justifies the 2nd statement in our proof? ivision Property of Equality Transitive Property of Equality Reflexive Property of ongruence efinition of ilation Slide 230 / Since we know that ' = k & ' = k, what should be the third statement in our proof? The image of point is itself. That is, ' =. '/ = k '/ = k and only Slide 231 / What reason justifies the 3rd statement in our proof? ivision Property of Equality Transitive Property of Equality Reflexive Property of ongruence efinition of ilation Slide 232 / ecause we know that '/ = k & '/ = k, what can we say as our 4th statement? k = k '/ = '/ Δ ~ Δ'' Slide 233 / What reason justifies the 4th statement in our proof? ivision Property of Equality Transitive Property of Equality Reflexive Property of ongruence Substitution Property of Equality Slide 234 / So far, we have 2 sides that are proportional. What can we add as our 5th statement? k = k '/ = '/ Δ ~ Δ''

40 Slide 235 / What reason justifies the 5th statement in our proof? ivision Property of Equality Transitive Property of Equality Reflexive Property of ongruence Substitution Property of Equality Slide 236 / Since we have 2 sides that are proportional and a pair of angles that are congruent. What can we add as our 6th statement? ''/ = k '/ = '/ = ''/ Δ ~ Δ'' Slide 237 / What reason justifies the 6th statement in our proof? Side-Side-Side Similarity ngle-ngle Similarity Side-ngle-Side Similarity efinition of Similar Polygons Slide 238 / If we know that the two triangle are similar, what can we add as our 7th statement? ''/ = k '/ = '/ = ''/ '' = k Slide 239 / What reason justifies the 7th statement in our proof? efinition of a ilation Multiplication Property of Equality ivision Property of Equality efinition of Similar Polygons Slide 240 / If we know that '/ = '/ = ''/ and an earlier statement showing proportions, what can we add as our 8th statement? ''/ = k '/ = '/ = ''/ '' = k

41 Slide 241 / What reason justifies the 8th statement in our proof? Substitution Property of Equality Multiplication Property of Equality ivision Property of Equality Transitive Property of Equality Slide 242 / If we know that ''/ = k, what can we add as our 9th statement? ''/ = k '/ = '/ = ''/ '' = k Slide 243 / What reason justifies the 9th statement in our proof? Substitution Property of Equality Multiplication Property of Equality ivision Property of Equality Transitive Property of Equality Slide 245 / 273 Slide 244 / 273 elow is a completed version of the proof that we just wrote. Given: '' is the image of after a dilation centered at point and with a scale factor k, k > 0. Prove: '' = k Statements Reasons 1) '' is the image of after a 1) Given dilation centered at point and with a scale factor k, k > 0. 2) ' = k ; ' = k 2) efinition of a ilation 3) '/ = k; '/ = k 3) ivision Property of Eq. 4) '/ = '/ 4) Transitive or Substitution Property of Equality 5) 5) Reflexive Property of 6) Δ ~ Δ'' 6) SS ~ 7) ''/ = '/ = '/ 7) efinition of Similar Polygons 8) ''/ = k 8) Transitive or Substitution Property of Equality 9) '' = k 9) Multiplication Property of Equality Slide 246 / 273 Warm Up 1. hoose the correct choice to complete the sentence. Rigid motions and dilations both preserve angle measure / distance. Similarity Transformations 2. omplete the sentence by filling in the blanks. rigid motions dilations preserve distance; do not preserve distance. 3. efine similar polygons on the lines below. Return to Table of ontents

42 Slide 247 / 273 Triangle GHI has vertices G(-4, 2), H(-3, -3) and N(-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. rawing Transformations Slide 248 / 273 rawing Transformations Δ LMN has vertices L(0, 2), M(2, 2), and N(0, 1). For each similarity transformation, draw the image R x-axis(δlmn ) Step 2 T <4, 2> (ΔGHI) G' (, ) H' (, ) I' (, ) Step 3 2(ΔG'H'I') G" (, ) H"(, ) I" (, ) Slide 249 / 273 rawing Transformations ΔLMN has vertices L(0, 2), M(2, 2), and N(0, 1). For each similarity transformation, draw the image. Slide 250 / 273 escribing Transformations What is a composition of transformations that maps trapezoid onto trapezoid MNHP? 2. 2 r (270, O)(ΔLMN ) Slide 251 / 273 escribing Transformations For each graph, describe the composition of transformations that maps Δ onto ΔFGH 1. Slide 252 / 273 escribing Transformations For each graph, describe the composition of transformations that maps Δ onto ΔFGH G F G H F H

43 Rotate quadrilateral 90 counterclockwise ilate it by a scale factor of 2/3 Translate it so nswer that vertices and M coincide. ~ MNPQ Slide 253 / 273 Similar Figures Two figures are similar 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. 9 N 4 P in. in. 6 in. 6 in. M Q N M Rotate ΔLMN 180 so that the vertical angles coincide. ilate it by nswer some scale factor x so that MN and JK coincide. ΔLMN ~ ΔLJK L J K Slide 254 / Which similarity transformation maps Δ to ΔEF? Slide 255 / Which similarity transformation does not map ΔPQR to ΔSTU? Slide 256 / Which of the following best describes a similarity transformation that maps ΔJKP to ΔLMP? M J a dilation only P L K a rotation followed by a dilation a reflection followed by a dilation a translation followed by a dilation Slide 257 / Triangle KLM is the pre-image of ΔK'L'M', before a transformation. etermine if these two figures are similar. Slide 258 / 273 PR Sample Questions Select from the choices below to correctly complete the sentence. Triangle KLM similar to ΔK'L'M', is is not which we can determine by a. dilation of scale factor -0.5 centered at the origin dilation of scale factor 1 centered at the origin E dilation of scale factor 1.5 centered at the origin F translated left 0.5 and up 1.5 G translated left 1.5 and up 0.5 PR Released Question - P - Non-alculator - Question #5 The remaining slides in this presentation contain questions from the PR Sample Test. fter finishing this unit, you should be able to answer these questions. Good Luck! Return to Table of ontents

44 Slide 259 / 273 Slide 260 / 273 Question 2/7 Topic: Identifying Symmetry w/ Transformations Question 7/7 Topic: ilation a. (0,0) d. 2 b. (1,-2) e. 3 c. (-2,-1) f. 4 Question 4/25 PR Released Question - EOY Slide 261 / 273 Topic: Rotation PR Released Question - EOY Slide 262 / 273 Question 6/25 Topic: ilation a. b. c. a. b. c. a. parallel to b. perpendicular to c. the same line as PR Released Question - EOY Slide 263 / 273 PR Released Question - EOY Slide 264 / 273 Question 9/25 Topic: ilation Question 12/25 Topic: omposition of Transformations PR Released Question - EOY PR Released Question - EOY

45 Slide 265 / 273 Slide 266 / 273 Question 14/25 Topic: Reflection Question 14/25 Topic: Reflection PR Released Question - EOY Slide 267 / 273 PR Released Question - EOY Slide 268 / 273 Question 21/25 Topic: omposition of Transformations Question 21/25 Topic: omposition of Transformations PR Released Question - EOY PR Released Question - EOY Slide 269 / 273 Slide 270 / 273 Question 21/25 Topic: omposition of Transformations Question 3/7 Topic: ilation PR Released Question - EOY PR Released Question - P

46 Slide 271 / 273 Slide 272 / 273 Question 5/7 Topic: Similarity Transformations Question 4/11 Topic: ilation a. b. a. b. c. d. e. a. is b. is not PR Released Question - P a. dilation of scale factor -0.5 centered at the origin b. dilation of scale factor 1 centered at the origin c. dilation of scale factor 1.5 centered at the origin d. translation left 0.5 and up 1.5 e. translation left 1.5 and up 0.5 Slide 273 / 273 PR Released Question - P Question 9/11 Topic: ilation PR Released Question - P