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Transformations at the Beach By Daniella Kay, Lucy Almberg, Greg Lubin, and James Kim Table Of Contents ● ● ● ● ● Reflections- Slide 3 Dilations- Slide 13 Translations- Slide 19 Tessellations- Slide 24 Rotations- Slide 26 Reflections a transformation which uses a line that acts as a mirror to reflect an image in that line. Reflections at the Beach ● When building a sand castle, you reflect the pail onto the sand. ● When you are riding in a boat, you can see the reflection of the boat in the water Key Vocabulary ● Line of Reflection- the line which acts like a mirror in a reflection ● Line of Symmetry- a line in a figure where the figure can be mapped onto itself by a reflection in the line Determining Lines of Symmetry ● All regular polygons have the same number of lines of symmetry as they do sides Lines of Symmetry at the Beach Beach towels are rectangles and have 1 line of have 2 lines of symmetry Sunglasses symmetry Reflecting a Preimage over a Line ● Reflections are an isometry so the size of the preimage stays the same when reflected ● Each point of the image will be the same distance from the line of reflection as the preimage, just on the opposite side Rules for Reflecting Over Specific Lines ● ● ● ● Rx-axis (x,y) → (x, -y) Ry-axis (x,y) → (-x,y) Ry=x (x,y) → (y,x) Ry= -x (x,y) → (-y, -x) A(-2,4) D(-2,1) B(4, 4) C(4, 1) Rx-axis D’(-2,-1) C’(4,-1) A’(-2,-4) B’(4,-4) Finding Equations for Lines of Reflection ● Find the midpoints between the points of the image and preimage ● Use this two points and the slope formula to find the slope of the line of reflection ● Plug the slope into the equation y=mx+b with one of the points to solve for b, which is the y-intercept of the line of reflection Finding Minimum Distance ● Find point C on the x-axis so AC+BC is a minimum distance ● To find point C, reflect either point A or B over the x-axis ● Draw a line from the new point (A’ or B’) to the point that was not reflected ● The point where the line intersects the xaxis is Point C. Let’s Try It! Johnny owns a surf shop on an island with two beaches. He wants his surf shop to be an equal distance from each beach. Use minimum distance to help Johnny decide where on the road (line m) he should put his shop. A B m Dilations A dilation with the center C and scale factor K is a transformation that maps every point P in the plane to a point P’ so that the following properties are true. 1. If P is not the center point C, then the image P’ lies on CP. The scale factor k is a positive number such that CP’ k= CP and k =1 2. If P is the center point C, then P = P’ Dilations ● A dilation is not an isometry, the distance is not preserved. ● Dilations are the same shape but not the same size ● All sides must increase by the same scale factor Dilations Key words● reduction- when the scale factor is less than 1 but greater than 0, the dilation is a reduction ● enlargement- when the scale factor is greater than 1, the dilation is a enlargement What is the scale factor? Is it an enlargement or reduction? 8 ft 4 ft 3 ft 6 ft answer= 2 enlargement Scale Factor= 2 What are the coordinates of the image? pre-image Is this a dilation? 4 ft 12 ft answer- no 7 ft 3 ft Translations Vocabulary: Translation - A transformation that maps every two points so that the preimage and image of the segment made by connecting them for both points are parallel, collinear and congruent Initial point - the starting point of the the vector of a translation Terminal point - the ending point of a vector at the image Component form - A form of a vector that combines the horizontal and vertical components Vectors and Coordinate Notation A vector can be used to describe a translation by mapping out the distance of points from the starting point to the terminal point with horizontal and vertical distance. So if point A was (1,4) and the image of point A was (3,6) the vector form of the translation would be <2,2> because the point moved two units to the right on the x-axis and 2 units up on the y-axis. Coordinate notation is another way of describing a translation. For example, if we use point A (1,4) and point A’ (3,6), the way to describe that translation with coordinate notation would be (x + 2, y+2). Matrices Matrices are a different way to describe transformations, points, and shapes in a coordinate plane. For point A (4,5), in matrix form it would be [ 4 ] for describing a translation [ 2 ]. 5 2 The matrix for the translation means two units right and two units up. To figure out the image of the translation you would add them. So [ 4 ] + [ 2 ] would be [ 6 ] 5 2 7 How to See if a Transformation is a Translation All translations are isometries, that means that the preimage and the image of the polygon, or segment that is translated will be congruent. Not a Translation Is a Translation The segments that are made when connecting the preimage to the image for the points in a translation are always parallel and congruent to each other. Real World Applications In the real world, vectors and coordinate notation are used to describe movement with coordinates. For example: The boat at the shore was at 39 degrees and 55 minutes North and 74 degrees and 4 minutes West (-74.07, 39.92) and the boat at the ocean was at 39 degrees and 53 minutes North and 74 and 3 minutes West (-74.05, 39.88) Can you write in vector form and coordinate notation the translation? Tesselations Vocabulary: Frieze pattern (or border pattern) - a pattern that extends to the left and right in a way so that a pattern can be mapped onto itself with horizontal translation. Types of Frieze patterns: T: Translation TR: Translation and 180 degree rotation TG Translation and horizontal glide reflection TV: Translation and vertical line reflection TRVG: Translation, 180 degree rotation, vertical line reflection and horizontal glide reflection TRHVG: Translation, 180 degree rotation, horizontal glide reflection, vertical line reflection Frieze Patterns A frieze pattern with horizontal translation: A frieze pattern with horizontal translation and 180 degree rotation: Rotations A rotation is a transformation in which a figure is turned about a fixed point Rotations at the Beach ● When you put an umbrella up by turning it in the sand, the top piece of the umbrella rotates. ● When you are swimming in the ocean and do a flip, you are rotating 360°. Rotations Key Vocabulary: ● Center of rotation- the fixed point around which the preimage rotates ● Angle of rotation- Rays drawn from the center of rotation to a point and its image from the angle at which the figure rotated ● Rotational Symmetry- when a figure can be mapped onto itself by a clockwise rotation of 180º or less ● Theorem 7.2- A rotation is an isometry Determining if a figure has rotational symmetry ● All regular figures have rotational symmetry ● If a figure is not regular, it could still have rotational symmetry. In order to find out if it has rotational symmetry, find the number of lines of symmetry in that figure and determine if you would be able to rotate it less than 360º so that the image is the same as the preimage. Do these figures have rotational symmetry? Yes No Determining the Angle of Rotation ● For a regular figure, divide 360 by the number of sides the figure has. ● For an irregular figure, you can divide 360 by the number of lines of symmetry it has. The starfish has 5 sides 360/5=72 The angle of rotation of the starfish is 72º. Drawing a rotation image using a protractor Rotate ABC 140° counter clockwise about point K. A 1. Draw a segment from vertex A to K point K. B 2. Measure a 140° angle using KA as one side and mark it. 3. Measure the length of KA. 4. Draw a segment the same length as KA, going from point K towards the 140° mark and make that point A’. 5. Erase all the lines you drew but keep point A’. 6. Repeat steps 1-5 for vertices B and C and then draw C Using a Protractor Cont’d ° 140 A’ 3cm 3cm A K B C Rotating a Figure 90º,180º,and 270º Just use these equations to rotate the figure: Clockwise: Counter Clockwise: R - (x,y) = (y, -x) R - (x,y) = (-x, -y) R - (x,y) = (-y, x) 90 180 270 R - (x,y) = (-y, x) R - (x,y) = (-x, -y) R - (x,y) = (y,-x) 90 180 270 Find the new Coordinates! Rotate the figure 180º clockwise. Coordinates of the Preimage: A (5,3) B (2,3) C ( 5,-1) D (2, -1) Coordinates of the Image: A’ (-5,-3) B’ (-2,-3) C’ (-5,1) D’ (-2,1) Finding the Angle of Rotation when a preimage is reflected over two lines ● ● If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is a rotation about point P. The angle of rotation is 2x°, where x° is the measure of the acute or right angle formed by k and m. Lets try it! What is the angle of rotation? Image Final Image 160° x=80 2x=160 80° Preimage T R A N S F O C V R O O C S A S ● ● ● ● ● ● ● ● ● ● Bibliography http://www.google.com/url?sa=i&rct=j&q=lines+of+symmetry&source=images&cd=&cad=rja&uact=8&docid=hBsHfOWLGH V_mM&tbnid=L8oFOQhlrgNmM:&ved=0CAQQjB0&url=http%3A%2F%2Fmymathhomework.jimdo.com%2Fgeometry%2Fnumber-of-lines-ofsymmetry%2F&ei=C4RnU5S2MeHC8QHPtoCoDg&bvm=bv.65788261,d.aWw&psig=AFQjCNFJaO8upkFDH_HZ8OWJIa U5VZa10A&ust=1399379232087287 http://www.moomoomath.com/regular_polygon_formula.JPG http://www.google.com/url?sa=i&rct=j&q=heart+sunglasses&source=images&cd=&cad=rja&uact=8&docid=u0p2HYaHgwz J3M&tbnid=C4t4D9oqHF608M:&ved=0CAQQjB0&url=http%3A%2F%2Fwww.toysrus.com%2Fbuy%2Fsunglasses%2Fkoa la-kids-valentines-day-heart-sunglasses-2247989428526556&ei=NtJnU8DGGumD8QHXyID4Bw&psig=AFQjCNHNrmPz1DoAq9WFIGrWgtaNhzfJg&ust=1399399024987425 http://www.google.com/url?sa=i&rct=j&q=beach%20pail&source=images&cd=&cad=rja&uact=8&docid=xmDNa0XYWZAW QM&tbnid=-I0M2ZkVJvBKsM:&ved=0CAQQjB0&url=http%3A%2F%2Fwww.walmart.com%2Fip%2FMy-Sunshine-Giant-2Gallon-Beach-PailShovel%2F16517591&ei=c9JnU6yTM6K88AGPkIDgAw&psig=AFQjCNE9oAthdSs4sB2EU05O10Z3Rowz7g&ust=139939 9408156961 http://www.google.com/url?sa=i&rct=j&q=boat%20reflecting%20in%20water&source=images&cd=&cad=rja&uact=8&docid =WJJpWJmLL66G7M&tbnid=JEI0auzDrT0LM:&ved=0CAQQjB0&url=http%3A%2F%2Fvisionmena.com%2FArkansas%2Fboating_law.htm&ei=ytNnU6QN6PK8AHzioCIDw&psig=AFQjCNEu2116WnUB0auaYet5JjycEbTZwg&ust=1399399691995375 http://www.pamsclipart.com/clipart_images/tropical_beach_with_a_sailboat_palm_tree_and_beach_chair_0515-10111202-2435_SM.jpg https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcRWOqCiIHnqt3aemmB_pnVcolSlon_5hFGBfrejlPIo00AS2zq8vg http://www.ilovebramhall.com/2014/04/jonathan-meare-beach-towel-clip/ http://www.mathsisfun.com/geometry/images/reflect-graph.gif http://www.cn-tvitem.com/upfile/20100407151950.jpg Bibliography Continued ● http://www.shindigz.com/party/beach-pail-set/pgp/9web0240 "Beach Arrow Facebook Covers for Timeline." 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