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Kayla Sampayo, Sean Carey, Jotaro Kurachi, Joanna Ha, and Sara Manzelli Translations: Joanna Rotations: Kayla Reflections: Sara Dilations: Jotaro Tessellations: Sean To put it simply, a translation is like a slide… Translation Definition PREIMAGE • It’s when you two images, the preimage and the image, in different locations but look exactly the same. You do it without rotating or flipping the image. It is an isometry. It’s like how a monkey moves from one tree to another. IMAGE Examples P P’ • 1) PP’ = QQ’ • 2) PP’ || QQ’, or PP’ and QQ’ are collinear. Q Q’ Double reflections • It’s when you reflect an image twice, then you’ll get a image that is transformed. reflecting translation Component form and vector Example • The component form of a vector combines the horizontal and vertical components. DEFINITION OF VECTOR: • A VECTOR IS A QUANTITY THAT HAS BOTH DIRECTION AND MAGNITUDE OR SIZE. COMPONENT FORM COMPONENT FORM Definition of component form: How to find vector component form • If you have line JK, and you want to find the component form, you create a right triangle with the line. K Then count how much from j to k it moves from the x and y axis. Then you have your component form. 4 3 2 1 J -6 -5 -4 -3 -2 -1 ANSWER: COMPONENT FORM OF JK IS <-6, 4> PROBLEMS! Find the component form of vector AB. B A <-9, 5> Equation to translation The equation for translation is: (x,y) (x+a, x+b) • a = horizontal change • B = vertical change So basically what your doing is adding the component form to a coordinate pair. Example: Find the new vertices after the transformation of parallelogram ABCD. The component form is <2, -4>. A D B C A (6,7); B (7,2); C (10,2); D (9,7) A’ – (6 + 2, 7 + -4) = (8, 3) B’ – (7 + 2, 2 + -4) = (9, -2) C’ – (10 + 2, 2 + -4) = (12, -2) D’ – (9 + 2, 7 + -4) = (11, 3) Practice problem! Find the coordinates of this shape after a translation. Find the component form using vector bc and the new coordinates. Then graph it. Use the equation! A C B • Answers: • Component form: <3, 2> • A’ = (3, 9) • B’ = (4, 6) • C’ = (7, 8) Matrices • Another way to get the coordinates of an image after translation is by using matrices. • Matrices is a group of numbers. • You but the x value on the top and the y value on the bottom in between two big brackets. • You get a second set of brackets and put in the component form and repeat the x and y depending on how many vertices there are. For example: If a triangle has three vertices, then you would repeat the component form three times. • Then you add the two brackets together according to their positions. If the component form was <5, -7>… Use matrices to find the translation of figure EFG. Use vector FG for the component form. G E F E’ = (3, 6) F’ = (7, 4) G’ = (8, 7) PRACTICE! Using matrices, find the coordinates after a translation, using the vector CD. A B D Answers: • Component form: <-1, 6> • A’ = (5, 13) • B’ = (4, 7) • C’ = (7, 7) • D’ = (6, 13) C A word problem A group of scientist researchers are planning to move their research center in a rainforest located in South America. They put the building on the coordinates A(1,2), B(2,6), C(4,7), and D(5,1). Find the component form of vector CD and find the transition from their original building to the new building. Use the equation or matrices to solve. B Component form: <-1, 5> A’ = (1 + -1, 2 + 5) (0,7) B’ = (2 + -1, 6 + 5) (1,11) C’ = (4 + -1, 7 + 5) (3,12) D’ = (5 + -1, 1 + 5) (4,6) C A D Translations are everywhere in the world. In the rainforest, translations are seen all the time when animals are walking, flying, swimming, climbing, and crawling! This Jaguar is stalking its prey! WORD SCRAMBLE CRTOVE 7 CNNETPOMOMORF 13 6 14 4 NTLRAINSAOT 3 10 ______ SICMTARE 5 8 DBLEUONTRFELEOTIC 1 11 GEPMIEAR 12 2 GEMIA 9 ________ ________ _____ ___________ ______ ___________ ________ _____ ______________ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 A rotation is a transformation in which a figure is turned about a fixed point. Some important vocabulary terms to help you better understand rotations are below: • Center of rotation- the fixed point of a rotation • Angle of rotation- rays drawn from the center of rotation to a point • Rotational symmetry- a figure in a plane that can be mapped onto itself by a clockwise rotation of 180° or less Here is an example of a rotation. The center of rotation is point P One way to rotate a figure You can use these equations to plot the new point of the figure that is rotating 90°, 180°, or 270° • R90 (x,y)= (-y,x) • R180 (x,y)=(-x,-y) • R270 (x,y)= (y,-x) • R-90 (x,y)= (y,-x) Remember that R-90 refers to a rotation of 90° clockwise Another way to rotate • First, you must identity where the center of rotation is. • Once you locate it, draw a line from one of the vertices to the center of rotation. Keep in mind that the center of rotation can be inside a figure, outside the figure, or one of the vertices • Use a protractor and mark where the angle of rotation is. Make sure you measure the angle counterclockwise unless the problem says otherwise. • Measure the length of the line you drew in second step. • Draw a line from the center of rotation that is the same measurement as the line you drew in step two. The new line should meet with the mark you made from step 4. • The endpoint of the line in step 5 is the prime version of the vertex you selected in step 2 • Lastly, repeat the steps for each vertex and connect the dots to form the final figure. This diagram illustrates the steps to find one of the point on the final figure. In order to decide whether a figure has rotational symmetry… You have to see whether the figure can be mapped onto itself by a clockwise rotation of 180° or less. For example, both of these shapes have rotational symmetry. If the purple polygon is rotated 180° or 60°, it is perfectly aligned with the original image. If the other figure is turned 180° it will also map onto them original image. Theorem 7.3 states… • If L1 and L2 intersect at point P, then a reflection in L1 followed by a reflection in L2 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 L1 and L2. Important Concept for Rotation Sample Problems Which of these figures rotational symmetry? Answer: The recycling logo What is the angle of rotation? Answer: 100 degrees When the parrot changes course it rotates around his right wing. This wing acts as the point of rotation. The parrot rotates 90° clockwise. Rotation Activity! Rotate the sloth 180° clockwise on the tree branch It should look something like this: • A reflection is a transformation which uses a line that acts like a mirror, with an image reflected in the line. • The line which acts like a mirror in a reflection is called the line of reflection. • A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line. • A reflection is an isometry, the original figure and the image are congruent Let’s learn about Reflections When reflecting a point over the x-axis, y-axis, y=x, or y=-x just follow these formulas! How do you reflect a point over a line such as x= -2? The reflected point will be the same distance away from the line of reflection. If easily seen, count how far away the point is from the line, it will be the same distance away from the line when reflected across it (1 to the left, it will then be 1 to the right) Reflect the triangle over the y-axis • Find the new points by using the formula: (x,y) = (-x,y) • The new points should be: (1,2) (1,6) (4,2) • Plot the new points to form the image. 7 (-1,6) 6 5 4 3 (-4,2) -4 -3 -2 (-1,2) 2 -1 1 0 1 2 3 4 A figure has a line of symmetry if it can be mapped onto itself by a reflection. The number of lines of symmetry in a regular polygon will be equal to the number of vertices • How many lines of symmetry do these shapes have? Regular Pentagon has 5 lines of symmetry Rectangle has 2 lines of symmetry Isosceles Trapezoid has 1 line of symmetry There are examples of reflections everywhere in the world around us. The rainforest is home to many butterflies. These butterflies all have a line of symmetry. Using your knowledge of reflections and lines of symmetry, decorate your own butterfly! Dilations • A dilation is a transformation where the original image and the after image are • • • • the same shape but differ in size. The scale factor cannot be less than zero. The scale factor is represented as k. A reduction is when the scale factor is greater than zero but less than 1. An enlargement is when the scale factor is greater than 1. Here’s some important vocabulary! This would be a reduction, since the afterimage is smaller than the pre-image. When the center is the origin, you can find the coordinates of the afterimage by multiplying the coordinates of the pre-image by the scale factor The scale factor is 2. For example, the coordinates of the pre-image are: (1,1) (3,1) (2,3) Therefore, the coordinates of the afterimage would be: (2,2) (6,2) (4,6) You can also use matrices to solve for the coordinates of the afterimage. You put the numbers in a matrix, then you multiply them by the scale factor . x2 Now you try! Answer: This is an enlargement because the image is larger than the pre-image. The scale factor is 3 When a tree is cut down, you can see the tree rings. Tree rings are used to determine how old the tree is. These rings are dilations of each other. Each outer ring is an enlargement of its inner rings. Believe or not, there are dilations all around us, even inside trees! What is a Tessellation? A tessellation is a repeating pattern of figures that completely covers a plane without any overlaps or gaps You can make a tessellation only by using isometrics (Translation, rotations, and reflections) Triangles and Quadrilaterals can always tessellate D C What is an easy way to tell if a figure can be tessellated? Do you remember the equation (n-2)180/n? If 360 is divisible by your answer, then it will tessellate. Tessellation Vocab • Vector- a quantity that has both direction and magnitude, it is represented by an arrow drawn between two points. • Transformation- The operation that maps or moves a pre-image onto an image • Regular Polygon- a polygon that is equiangular and equilateral • Semi regular tessellation- more than one kind of regular polygon is used and the same arrangement of polygons meets at any vertex of the tessellation. Here are two examples of semi regular tessellations: Try to Tessellate! Can you tessellate this caterpillar? Your tessellation should look like this: Tessellations in the Rainforest There are tessellation all around us in our world and especially in the rain forest! A perfect example of a real world example of a tessellation is a turtle shell, you may not realize but a turtle shell is a tessellation! Bibliography: Websites • http://office.microsoft.com/en-us/powerpoint-help/animate-text-or-objectsHA010021497.aspx • http://www.mathsisfun.com/algebra/vectors.html • http://www.mathsisfun.com/algebra/matrix-introduction.html • http://www.tessellations.org/tessellations-all-around-us.shtml • http://puzzlemaker.discoveryeducation.com/WordSearchSetupForm.asp?campaign=fly out_teachers_puzzle_wordcross • http://www.virtualnerd.com/middle-math/integers-coordinateplane/transformations/reflection-definition • http://mathbydesign.thinkport.org/images/pdfs/TransformationsReflections_LessonPla n.pdf • http://www.regentsprep.org/Regents/math/geometry/GT1/reflect.htm Bibliography: Pictures • http://www.clker.com/cliparts/T/b/y/u/y/A/rain-drop-md.png • http://imgs.tuts.dragoart.com/how-to-draw-a-rainforest_1_000000004065_5.jpg • http://www.canvas101.co.uk/images/gallery-art/Animal/AN0072%20Large%20Red%20and%20Blue%20Macaw%20Parrot%20Flying%20.jpg • http://www.education.vic.gov.au/images/content/studentlearning/mathscontinuum/RotationalExamples.gif • http://o.quizlet.com/jS0zgaFgt4ZTZSacC2ABzQ_m.png • http://upload.wikimedia.org/wikipedia/commons/1/1e/Heart-SG2001-transparent.png • http://www.euclidraw.com/index_fls/Rotation.GIF • http://i1.wp.com/proofsfromthebook.com/wp-content/uploads/2013/07/reflection-rotation.png?resize=450%2C301 • https://img0.etsystatic.com/035/0/7332824/il_340x270.525017676_flc4.jpg • http://moodle.tbaisd.org/pluginfile.php/68845/mod_book/chapter/50821/dilation.jpg • http://moodle.tbaisd.org/pluginfile.php/68845/mod_book/chapter/50830/dilation%202.jpg • http://www.johnlwarren.net/images/159.jpg • http://www.webquest.hawaii.edu/kahihi/mathdictionary/images/lineofsymmetry.png • http://ppcdn.500px.org/19643125/dcf7b3e8417ccc5b5f4d6e508fe3332102db2b5f/5.jpg • http://www.layers-of-learning.com/wp-content/uploads/2012/11/butterfly.jpg