Name: Period: Unit 1. Modeling with Geometry: Transformations

Name: Period: Unit 1 Modeling with Geometry: Transformations 1 2017/2018

2 2017/2018

Unit Skills I know that... Transformations in general: A transformation is a change in the position, size, or shape of a geometric figure. (G-CO.2) There are two types of transformations, rigid and non-rigid motions. (G-CO.2) The pre-image is the original figure and the image is the transformed figure. (G-CO.2) Congruent figures have the same size and shape. (G-CO.2,4) Rigid motions preserve the size and shape (or distance and angle measure) of a figure. (They are sometimes called congruence motions.) (G-CO.2) Rigid motions include translations, reflections, and rotations. (G-CO.2) Translations: A translation is a transformation where a figure slides without turning. (G-CO.4) Lines that connect the corresponding points of a pre-image and its translated image are parallel. (G- CO.4) Corresponding segments of a pre-image and its translated image are parallel. (G-CO.4) A translation does not change the orientation of a figure. (G-CO.4) Reflections: A reflection is a transformation that flips a figure across a line, called the line of reflection. (G-CO.4) Segments connecting corresponding points of a pre-image and its reflected image are bisected by the line of reflection. (G-CO.4) Corresponding points of a pre-image and its reflected image are equidistant from the line of reflection. (G-CO.4) The reflection of a figure changes orientation so that it faces in the opposite direction of the original figure. (G-CO.4) Rotations: A rotation is a transformation where a figure turns around a fixed point, called the center of rotation. (G-CO.4) The amount of a rotation is described by the angle of rotation. The angle of rotation is typically described as part of a turn (e.g. ¼ turn) or in degrees. (G-CO.4) Corresponding points of a pre-image and its rotated image fall on concentric circles whose center is the center of rotation. (G-CO.4) Lines that connect the corresponding points of a pre-image and its rotated image to the center of rotation form an angle equal to the angle of rotation. (G-CO.4) A rotation does not change the orientation of a figure. (G-CO.4) Dilations: A dilation is a transformation where a figure is reduced or enlarged by a given scale factor with respect to a point called the center of dilation. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. (G-SRT.1) A dilation is not a rigid motion. However, a dilation with a scale factor of 1 (or -1) produces congruent figures. (A scale factor of 1 produces the same figure on top of itself, while a scale factor of -1 produces the same figure rotated 180 around the center of dilation.) (G-SRT.1, G-CO.2) A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. In other words, corresponding segments of a pre-image and its dilated image are the same segment or parallel segments. (G-SRT.1) 3 2017/2018

Intro to Geometric Transformations Video Intro to Translations (math/geometry/hs-geo-transformations/hs-geo-transformations-intro/a/intro-to-translations) Practice Problems: None Transformation is taking one set of or a set of points, and then it into a set of coordinates or a different set of point. How many points are represented by the quadrilateral that is being transformed? List the types of rigid transformations: The Image is Rigid transformations maintains the and size. and are preserved. Where are transformations used in real life? Class Notes: The Preimage is Prime Notation: to distinguish the pre-image from the image, use a Input points Output points Non-rigid transformation is Example 4 2017/2018

Translations Videos: Performing Translations (/math/geometry/hs-geo-transformations/hs-geo-translations/v/drawing-image-of-translation) Practice Translating Points 1 2 3 Determining Translations (/math/geometry/hs-geo-transformations/hs-geo-translations/v/formal-translation-tool-example) Practice Determining Translations 1 2 3 Practice Translating Shapes 1 2 3 Translation Challenge Problem (/math/geometry/hs-geo-transformations/hs-geo-translations/v/determing-a-translation-between-points) Performing Translations What was the name of the triangle used in the translation? What was the x coordinate of Point I in the Image? Vector Notation: <x, y> the x is the horizontal translation and the y is the vertical translation Properties of Translations Line segments are taken to line segments of the same length Angles are taken to angle of the same measure Lines are taken to lines and parallel lines are taken to parallel lines Translating Shapes Translation Notation: T(8, -1) means every point is moved in the direction and in the direction Determining Translations What was the final translation that mapped WIN onto the other triangle? Translation Challenge Problem (complete the problem below from the video) A translation acting on the coordinate plane takes point (-169, 434) to point (-203, -68). What are the coordinates of the image of point (31, -529) under this translation? 5 2017/2018

Class Notes: Types of notations for translations Descriptive Notation: Example: Coordinate Notation (function rule): Example: T(x,y) (x+2, y-3) Vector Notation o A vector has and Example: 3 mph due north o denoted by <a,b> a specifies a change b specifies a change negative: move or T<2,-3> (x+2, y-3) 6 2017/2018

Classwork Translation of Shapes( 2011 Kuta Software LLC. All rights reserved.) Classwork Complete Khan Academy Quiz #1 https://www.khanacademy.org/math/geometry/hs-geo-transformations#quiz-1 7 2017/2018

Rotations Videos: Rotating shapes (/math/geometry/hs-geo-transformations/hs-geo-rotations/v/points-after-rotation) Practice Rotate Points 1 2 3 Determining Rotations (/math/geometry/hs-geo-transformations/hs-geo-rotations/v/defining-rotation-example) Practice Determining Rotations 1 2 3 4 Practice Rotating Shapes 1 2 3 4 Performing Rotations Positive degrees: rotates the image Negative degrees: rotates the image What order does Mr. Khan rotate the points of PIN in? The PIN with point P(2,3), I(7,-7) and N(2,-7) map to what points for a 90 rotation? P I N Determining Rotations What point are the line segments rotating around? What was the final angle of rotation? 8 2017/2018

Class Notes: A is a transformation that all points of a figure around a point called the (COR). The of rotation is the number of thorough which points rotate around the COR. Positive Negative A rotation is a transformation about a point P such that every point and its image are from P (lie on a ) all with vertex P formed by a point and its image have the same Notation R P, θ ( ABC) = A B C Rotation of 90 : R90 (x,y) Rotation of 180 : R90 (x,y) Rotation of 270 : R90 (x,y) 9 2017/2018

Rotations with calculator Figure being used will be a triangle: A(1,1), B(6,3), C(4,7). 1. Clear entries in the Y= positions (or turn them off). 2. Turn on the Connected Graph icon under StatPlots. Go to StatPlot - #1 Plot - highlight On - highlight second icon. Xlist is the name of the list where the x-coordinates will be found. Ylist is the name of the list where the y-coordinates will be found. Mark: choose the heavier mark to represent the original figure. 3. Enter values into L1 and L2. Enter the x-coordinate in L1 and its corresponding y-coordinate in L2. Enter the first coordinate again at the end to complete the connected drawing. The original triangle is now residing in Plot1. When graphing do NOT choose ZoomStat for the window. We want to have a coordinate axes for examining our transformations. Use a standard 10x10 window (ZStandard) for this problem. 4. We will be placing our transformation coordinates into L3 and our transformation figure will reside in Plot2. Rotation of 90 counterclockwise: (x, y) (-y, x) Store the negated y-values into list L3. You can type them in yourself or let the calculator create the values. Arrow up ONTO L3 and enter -L2 (to negate the y-values). Press ENTER. Rotation of 180 counterclockwise: (x, y) (-x, -y) Store the negated x-values into list L3 and the negated y-values in L4. Under Plot 2, assign Xlist: L3 and Ylist: Y4. Graph. Rotation of 270 counterclockwise: Try this on your own. 11 2017/2018

Create your own figure 1. Draw a figure in quadrant 1 with at least 5 sides 2. In the table, write the transformed coordinates for each rotation. Use a different color for each rotation. Plot the transformed shapes in the matching color on the coordinate plane. Point 90 180 270 Answer the following questions: For the 90 rotation, why did you plot L3 as the x-list and L1 as the y-list? For the 180 rotation, you had to create list Y4. Why did you create this list? Explain how you created the 270 rotation. 12 2017/2018

Reflections Videos: Reflecting Shapes (math/geometry/hs-geo-transformations/hs-geo-reflections/v/reflecting-segments-over-line) Practice Reflect Points 1 2 3 Determining Reflections (/math/geometry/hs-geo-transformations/hs-geo-reflections/v/points-on-line-of-reflection) Practice Determining Reflections 1 2 3 4 Practice Reflect Shapes 1 2 3 4 Reflecting Shapes In order to reflect a point, we need to create a line that is to the reflecting line and the points are the distance from the reflecting line. Lines that are perpendicular have slopes that are. Determining Reflections What does the reflection tool require to define a line? We need to calculate the to determine the line of reflection. 13 2017/2018

Class Notes: A is a transformation that all points of an image over a line called the (LOR). The LOR is the of each segment joining each point and its. Points are from the LOR. Reflection Notation: r k ( ABC) = Reflection over x-axis rx-axis= (x,y) Reflection over y-axis ry-axis= (x,y) Reflection over y = x or y = -x ry=x= (x,y) ry=-x= (x,y) Reflection over any line A reflection can occur across any line 14 2017/2018

Reflections across multiple lines Parallel Lines Theorem of Reflection 1. Using a ruler, measure from each point to line l to create the reflection A B C. Label the points. 2. Next, measure from each point to line m to create the reflection A B C. Label the points. 3. Create a line from A to A to A. Measure the of AA and AA. maa = maa = B l m C A Answer the following questions. What do you notice about AA and AA? How else could you write this transformation? 16 2017/2018

Intersecting Lines Theorem of Reflection This is called the 1. Using a ruler, measure from each point to line l to create the reflection A B C. Label the points. 2. Next, measure from each point to line m to create the reflection A B C. Label the points. 3. What is the angle of rotation from A to A? 4. What is the angle of rotation from A to A? m l B C A Answer the following questions. What do you notice about angle from AA and AA? How else could you write this transformation? Classwork Complete Khan Academy Quiz #2 https://www.khanacademy.org/math/geometry/hs-geo-transformations#quiz-2 17 2017/2018

Dilations Videos: Performing dilations ( /math/geometry/hs-geo-transformations/hs-geo-transformations-intro/v/dilating-from-an-arbitrary-point-example) Practice Dilate Points 1 2 3 4 Dilating Shapes: shrinking (/math/geometry/hs-geo-transformations/hs-geo-reflections/v/scaling-down-a-triangle-by-half) Dilating Shapes: expanding (/math/geometry/hs-geo-transformations/hs-geo-reflections/v/thinking-about-dilations) Practice Dilate Shapes 1 2 3 4 Determining Dilations (/math/geometry/hs-geo-transformations/hs-geo-reflections/v/dilating-one-line-onto-another) Practice Determine Dilations 1 2 3 4 Performing Dilations What point does point A map to? (, ) What point does point E map to? (, ) Dilating Shapes: shrinking What point does point D map to? (, ) What point does point E map to? (, ) What point does point F map to? (, ) Dilating Shapes: expanding What point does point B map to? (, ) AB is units long, and its image is units long. Determining Dilations In the first example, the center of dilation is on/not on (circle one) the line. Point (3, 3) would dilate to (, ) Therefore, the line will. In the second example, what is the center of dilation? (, ) What is the scale factor? 18 2017/2018

Class notes Dilations Dilation is a transformation that produces an image to the original by shrinking or stretching the of the pre-image. Similar images have shape and size. Scale Factor (k): of the lengths of the dimensions in similar images. (ie., sides or area) Notation: D k, O(x, y) = (kx, ky) k = A B AB, O is the center of dilation If O is a fixed point and A is the image of A, then 0, A and A are and k =. Enlarge: k Reduce: k Center of Dilation is the point about which all points are or. Given rectangle ABCD, find the dilation with a scale of 2/3. A(-6,-3) B(-6,3) C(6,3) D(6,-3) Determine the scale factor 19 2017/2018

Dilation from an arbitrary point Dilate the following k = 3 from point (10, -10) A(7,-8) B(5,-8) C(5,-3) D(9,-3) E(9,-5) F(7,-5) x distance from P to A: x distance from P to A : x value of A : y distance from P to A: y distance from P to A : y value of A : 20 2017/2018

Properties of Dilation Dilate ABC about the origin with a scale factor of 2. Graph the new triangle; label the vertices A, B, & C. B C A Complete the following using your dilation. Using a protractor, measure the angles of ABC and A B C. What do you notice? Using the distance formula, calculate the lengths of AB, A B, AC, A C, BC, and B C. What do you notice? What do you notice about AB and A? B AC and A? C Note that A and A lie on the origin. What conclusion can you make about the segments of an image when the corresponding segments of the preimage pass through the center of dilation? Using the slope formula, calculate the slopes of BC and B. C What do you notice? What conclusion can you make about the segments of an image when the corresponding segments of the preimage do not pass through the center of dilation? 21 2017/2018

2.6 cm 1.6 cm 2.5 cm Floor Plan Comparison Use these two-bedroom apartment floor plans to complete the tables and answer the questions below on your own paper. Apartment 1 Apartment 2 2.5 cm 2 nd Bedroom 2.3 cm 0.4 cm 2.8 cm 1.3 cm 1 st 1.2 cm 2.2 cm 2.5 cm 2.3 cm The scale factor for Apartment 1 is: 1 cm = 5 ft Apartment 1: Room Master Bedroom 2nd Bedroom Blueprint Dimensions (cm) Actual Dimensions Area The scale factor for Apartment 2 is: 1 cm = 4 ft Apartment 2: Room Blueprint Dimensions (cm) 1st Bedroom 2.5x 2.6 2nd Bedroom Actual Dimensions Area Living Room 2.1x2.1 Living Room Kitchen 1.5x 1.7 Kitchen Bathroom Bathroom 1.3x 1.6 Entry Way 1.2 x 1.7 Hall & Hall Closets 1.2x 2.0 Closet w/2nd BR 0.8x 0.5 Closet w/1st BR 1.3x 0.4 1. Calculate the total square footage of Apartment 1 and Apartment 2. 2. Apartment 1 rents for $550 per month. What is the price per square foot for Apartment 1? 3. Apartment 2 rents for $500 per month. What is the price per square foot for Apartment 2? 4. There are advantages and disadvantages to both apartments. Considering the various factors represented here in the plans and calculations, which apartment would you prefer to rent, and why? (There is not one correct choice here, but you should support your choice with multiple reasons using complete sentences and proper grammar and spelling.) 23 2017/2018

Composite Transformations When we perform two or more transformations on an image, it is called a COMPOSITE TRANSFORMATION. If we called the first transformation R and the second transformation P, then we have: R( ABC) = A B C and P(A B C ) = A B C We can put these together as a composition: P o R = P(R( ABC)) = A B C Example 1 Translate down 3, right 2 then reflect over the x-axis Write a rule for the first transformation (x, y) (, ) Use the results from the first translation and write a rule for the 2 nd transformation (, ) (, ) Proper form (, ) (, ) Example 2 Given triangle GHI with G(-2, 1), H(3, 4), and I(1, 5), find the points of the image under the following transformations and write the Algebraic Rule. Translate right 2, down 3 Reflect over the x-axis Rotate 90 degrees, counter-clockwise Dilate with a scale factor of 3 24 2017/2018

A glide reflection is the composition of a and a where the motion is to the. 1. Translate A B, then reflect over line m 2. Reflect over line m, then translate A B. A B A B m m 3. Does it matter which transformation is done first in a glide reflection? 4. Translate A B, then reflect over line m 5. Reflect over line m, then translate A B. B B A A m m 6. Is this a glide reflection? Why or why not? Practice: 1. The translation T ( x, y) ( x 2, y 4) is followed by the translation T ( x, y) ( x 8, y 5). Write this as a single translation. 2. The rotation R O,90 is followed by the rotation R O, 270. Write this as a single rotation. 3. The reflection R x axis is followed by the reflection R y axis. Can this be written as a single reflection? Can this be written as a single transformation? Draw a sketch to support your answer. 25 2017/2018

HW Composition of Transformations Part 1 All rectangles in the grid below are congruent. Follow the instructions and then write the number of the rectangle that matches the location of the final image. Rectangles may be used more than once. 7 1 5 6 2 4 3 8 Which rectangle is the final image of each transformation? Reflect Rectangle 1 over the y-axis. Then translate down three units and rotate 90 counterclockwise around the point (3, 1). (Hint: redraw the axes so that the origin corresponds to (3, 1).) Translate Rectangle 2 down one unit and reflect over the x-axis. Then reflect over the line x = 4. Reflect Rectangle 3 over the y-axis and then rotate 90 clockwise around the point (-2, 0). Finally, glide five units to the right. Rotate Rectangle 4 90 clockwise around the point (-3, 0). Reflect over the line y = 2 and then translate one unit left. Translate Rectangle 5 left five units. Rotate 90 clockwise around the point (-2, 2) and glide up two spaces. Rotate Rectangle 6 90 clockwise around the point (4, 4) and translate down three units. Rotate Rectangle 7 90 clockwise around (-4, 4) and reflect over the line x = -4. Reflect Rectangle 8 over the x-axis. Translate four units left and reflect over the line y = 1.5. 26 2017/2018

Composition of Motion Algebraic Rules Part 2 For each problem, there is a composition of motions. Using your algebraic rules, come up with a new rule after both transformations have taken place. If needed, you may use the coordinate plane below to help you determine the rule. 1) Translate a triangle 4 units right and 2 units up, and then reflect the triangle over the line y = x. 2) Rotate a triangle 90 degrees counter clockwise, and then dilate the figure by a scale factor of 3. 3) Translate a triangle 4 units left and 2 units down, and then reflect the triangle over the y-axis. 4) Rotate a triangle 90 degrees clockwise, and then dilate the figure by a scale factor of 1/3. 5) Translate a triangle 4 units right and 2 units down, and then reflect the triangle over the x-axis. 6) Rotate a triangle 180 degrees counter clockwise, and then dilate the figure by a scale factor of 2. 7) Translate a triangle 4 units left and 2 units up, and then reflect the triangle over the line y = x. 8) Rotate a triangle 180 degrees clockwise, and then dilate the figure by a scale factor of 1/2. 27 2017/2018

Warm-Up: Kim and Jim are twins and live at the same home. They each walk to school along the same path at exactly the same speed. However, Jim likes to arrive at school early and Kim is happy to arrive 7 minutes later, just as the bell rings. Pictured at right is a graph of Jim s distance from school over time. Interpreting Functions 1. Use a dotted line to sketch Kim s graph of distance from school over time (once she leaves for school). 2. How many minutes after 7AM does Jim leave for school? 3. How many minutes after 7AM does Jim arrive at school? 4. How many minutes after 7AM does Kim leave for school? 5. How many minutes after 7AM does Kim arrive at school? 6. What is Jim s farthest distance from school? 7. What is Jim s closest distance to school? 8. What is Kim s farthest distance from school? 9. What is Kim s closest distance to school? Use your answers to the above questions to fill in the following: Jim s domain: x (where x represents time after 7AM) Jim s range: y (where y represents distance from school) Kim s domain: x (where x represents time after 7AM) Kim s range: y (where y represents distance from school) Extend: Kim s graph is a horizontal translation of Jim s graph. When a graph translates horizontally, how do the domain and range change? 28 2017/2018

Domain and Range in translations Quick review: The domain is the set of all possible x-values on the graph. The range is the set of all possible y-values on the graph. Side note about notation: ** Discrete values must be represented by a list of values written in this notation: { 1, 5, 7} ** Interval notation: Represents the domain and range with a pair of numbers. [ ] for inclusive ( or ) and ( ) are used for exclusive (< or >). Example: 3<x 12 is equivalent to (3, 12] Note: - and are always exclusive 1. Describe the translation(s) from the pre-image to the image. a. Given the following graph, state the domain and range of the pre-image: Domain: Range:. b. State the domain and range of the image: Domain: Range:. 2. Draw and label the image of AB translated left 2 and down 3. a. State the domain and range of the pre-image: Domain: Range:. b. State the domain and range of the image: Domain: Range:. 3. Draw and label the image of AB reflected over the x-axis. a. State the domain and range of the pre-image: Domain: Range:. b. State the domain and range of the image: Domain: Range:. 29 2017/2018

4. Draw and label the image of AB reflected over the y-axis a. State the domain and range of the pre-image: Domain: Range:. b. State the domain and range of the image: Domain: Range:. 5. Draw and label the image of AB reflected over the line y = x. a. State the domain and range of the pre-image: Domain: Range:. b. State the domain and range of the image: Domain: Range:. 6. Draw and label the image of AB rotated 90. a. State the domain and range of the pre-image: Domain: Range:. b. State the domain and range of the image: Domain: Range:. 7. Draw and label the image of AB dilated by a factor of 3 with a center of (0,0) a. State the domain and range of the pre-image: Domain: Range:. b. State the domain and range of the image: Domain: Range:. 30 2017/2018

Homework Domain & Range Given the patterns seen above, can you predict the domain/range of an image given a pre-image domain/range? Let s try: 1. Given a relation composed of points A(2,5), B(1, -6), and C(4, 7), a. State the domain and range of the relation: D: { } R: { } b. State the domain and range of the image when the relation is: i. Translated right 2 and down 3 : D: { } R: { } ii. Reflected in the x-axis: D: { } R: { } iii. Reflected in the y-axis: D: { } R: { } iv. Reflected in the line y=x: D: { } R: { } v. Rotated 90 : D: { } R: { } vi. Dilated by a factor of 7 with a center of (0, 0): D: { } R: { } 2. Given a line segment with endpoints (0,4) inclusive and (3,0) exclusive a. State the domain and range of the segment. D: R: b. State the domain and range of the image when the relation is: i. Translated right 2 and down 3 : iv. Reflected in the line y=x: D: D: R: R: ii. Reflected in the x-axis: D: R: iii. Reflected in the y-axis: D: R: v. Rotated 90 : D: R: vi. Dilated by a factor of 7 with a center of (0, 0): D: R: 31 2017/2018

Basic Transformations and Algebraic Rules Translations (Slide) Function Rule (x, y) Vector Rule Picture Rotations 90 rotation (x, y) (counter-clockwise) 180 rotation (same as a reflection (x, y) around both axes) 270 rotation (x, y) (counter-clockwise) 360 rotation (x, y) Picture Reflection (Flip) Reflect over x-axis (x, y) Reflect over y-axis Reflect over both axes (same as a 180 rotation) Reflect over y = x Reflect over y = -x Picture (x, y) (x, y) (x, y) (x, y) Dilation (By a factor a, centered on the origin) a>1, (x, y) 0<a<1, Picture 32 2017/2018

Unit 1A: Modeling with Geometric Transformations Unit 1A Review #1-9, write the algebraic (arrow) rule for each transformation and the image points. Use graph paper if necessary. 1. MNO with M(-5,2), N(0, 4), and O(4, 5); translate left 6 and down 2. 2. Pentagon PENTA with P(0, 2), E(4,6), N(8, -1), T(6, -3), and A(2, -4); reflect across y-axis. 3. Rectangle ABCD with A(0,0), B(0,4), C(5,4), and D(5,0); rotate 90 clockwise. 4. Pentagon MNOPQ with M(-4, 1), N(-2, 3), O(0, 3), P(4, 3), and Q(2, -7); rotate 90 counterclockwise, then dilate by d = 4. 5. Translate ABC A(3, 6), B(4,2), C(5, 6) right 4 and up 3 units. Then reflect the triangle across the y-axis. 6. You are given ABC with A(1,4) B(-6,-2) C(4, -3). What rule transforms the triangle to give you the points A (1, -4) B (-6, 2) C (4,3)? 7. You are given pentagon JAROD with J(2, 0) A(4,2) R(6,0) O(6, -2) D(2,-2). What rule transforms the pentagon to the points J (0,-2) A (2, -4) R (0, -6) O (-2, -6) D (-2, -2)? 8. Perform these transformations in order on quad ABCD. A( -1, 1) B(0, 5) C(1, 2) D( 1, -1) (x, y) (3x, 3y) (x, y) (y, x) (x, y) (-x, -y) 33

Unit 1A: Modeling with Geometric Transformations #9-14 What are the final coordinates? 9. If translation T: (5, 3) ( 4, 0), then T: (8, 2) (, ) 10. T: (x, y) (x 5, y + 2), if F (7, 6), find F. 11. M is reflected over the y-axis. If M is (6, 1), find M. 12. C is rotated about the origin 90. If C is ( 9, 5), find C. 13. Y is rotated about the origin 180. If the image of Y is (0, -3) find Y. 14. A figure is reflected over the line y = x. If the preimage is (2, 7), find the image. 15. ABC has vertices A(5, 2), B( 4, 0), C(7, 1). Find the coordinates of the image of the triangle if it is dilated using the rule: D O,3. 16. Dilate ABC about point O using magnitude 1 4. A (, ), B (, ), C (, ) 17. DO, 2 (ABCD) = A B C D. The lengths of the segments of the preimage are as follows: AB = 6, BC = 5, CD = 3, AD = 4 a. What is the length of B C? A' A O D B C B' C' b. What is the length of? A B D' c. If the slope of CD is 1/3, what is the slope of C D? What allows you to make this conclusion? d. Why is the image of AB on the same line as AB? 34

Unit 1A: Modeling with Geometric Transformations 18. PQRST ~ UWXYZ with a scale factor of 2:5. If the perimeter of UWXYZ is 40 inches, what is the perimeter of PQRST? 19. A figure is reflected consecutively across two lines that are parallel and 12 cm apart. Describe the resulting transformation. Be specific. 20. A figure is reflected consecutively across two lines that intersect to form a 45 angle. Describe the resulting transformation. Be specific. 21. A figure is translated using the rule <6, 0> and then reflected in the y-axis. Is this composition of transformations a glide reflection? Explain why or why not. 22. For each problem, there is a composition of motions. Using your algebraic rules, come up with a new rule after both transformations have taken place. a. Translate a triangle 5 units left and 3 units up, and then reflect the triangle over the x-axis. b. Rotate a triangle 90 degrees counter clockwise, and then reflect in the line y = x. c. Reflect in the line y = x, and then translate right 4 units and down 2 units. 35

Unit 1A: Modeling with Geometric Transformations 36