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Transformations By Yena Shin, Nicola Licata, Mona Moshet & Andi Deng Table of Contents O Tessellations O Reflections by Mona Moshet O Dilations by Nicola O Rotation by Yena Shin O Translation by Andi Deng O Bibliography Tessellations Vocabulary • Tessellation- a repeating pattern of figures that Completely covers a plane without any gaps or overlaps • Edge- the intersection between two bordering tiles • Vertex- the intersection of three or more bordering tiles • Regular tessellation- when a tessellation uses only one type of regular polygon to fill up a plane • Semi-regular tessellation- when a tessellation uses more than one type of regular polygon to fill up a plane • Translational symmetry-when a translation maps that tessellation onto itself Vocabulary (continued) • Glide reflectional symmetry-when a glide reflection maps the tessellation onto itself • Reflection or line symmetry-when a figure is reflected across the axis & the image is the same as the original • Rotational symmetry-when a rotation of 180 degrees or less is performed on a tessellation and the resulting image is the same as the original image • Point symmetry-when a tessellation rotates 180 degrees and the image is the same Determining which polygons to tessellate or not • To see which regular polygons can tessellate or not you use the following formula to find the polygon’s angle measure • a=180(n-2)/n • If the angle measure is a factor of 360, then that polygon can tessellate • For example, if you wanted to see if a regular pentagon, like the one • at the right, could tessellate, you use the formula to find that each inner angle is 108 degrees • Then you would divide 360 By 108 360/108=3 1/3, therefore a regular pentagon cannot tessellate Tessellations Activity • Johnson wants to tile the floor of his house. However he wants to semi-tessellate the tiles with 3 different shapes. Can you help Johnson pick which shapes he needs? Reflections By Mona Moshet Vocabulary Line of symmetry • Reflecting Figures & Finding the Line of Reflection Rules for reflecting figures: Finding the Minimum Distance Activity • Take the paper that you have and bring the top left tip to the right side of the paper till a triangle is made. • Then fold the small rectangle under the triangle. You now have a square! • Fold the square in different ways to find how many lines of symmetry the square has • How many lines of symmetry does a square have? • This proves that a regular polygon’s number of lines of symmetry is equal to it’s number of sides Dilations Nicola Licata What is a Dilation? • A dilation is a transformation that reduces or enlarges a polygon by a given scale factor around a given center point. • When a figure is dilated the new figure will always be similar to the original figure Center Point There is a scale factor of 2 which means that the new figure is 2 times larger The Scale Factor • The scale factor is the amount by which the image grows or shrinks • To find the scale factor of two given shapes simply find two corresponding sides or points and put the new shape’s side over the original shape’s corresponding side. 15 Here the scale factor is 3 because 15 is the new distance and 5 is the old distance. 15/5 = 3 5 Reductions and Enlargements • If the scale factor of your dilation is greater than 1 than the dilation is an enlargement. • If your scale factor is greater than 0 but less than 1 than the dilation is a reduction This is an enlargement because the scale factor is 5. 20 100 Activity • Johnny wants to make a scale model of the clock tower big ben in London. If he wants to make it 1/100 of the actual height how tall will the scale model be? 316 ft. Rotation Key Vocabulary Rotation A transformation where a figure is turned around a center of rotation. Isometry A transformation where the figure stays congruent Center of Rotation A fixed point anywhere that the figure rotates about. The center of rotation can be anywhere. It can be inside or outside the figure. Angle of Rotation The angle created by rays drawn from the center of rotation to a point and its image. Key Concepts Theorem Line K and line M intersect at point P. Then a reflection in line K and the line M is a rotation about point P. The angle of rotation is double the angle formed by K and M. Since the angle formed by lines Kand M is 70°, the angle of rotation is 140°. Key Concepts Cont. Rotational Symmetry A figure in the plane has rotational symmetry if the figure can be mapped onto itself by clockwise rotation of 180 degrees or less. Equations: R90 (X,Y) = (-Y,X) R180 (x,y) =( -x,-y) R270 (x,y) =(y,-x) R-90 (x,y) = (y, -x) Rotational Symmetry Does this figure have rotational symmetry? If so, what is the angle of rotation? Rotating on a Coordinate Plane You can rotate the building by using the equations from the previous slide. You are to rotate counter clock wise unless told by the problem. Equations: R90 (X,Y) = (-Y,X) R180 (x,y) =( -x,-y) R270 (x,y) =(y,-x) R-90 (x,y) = (y, -x) Real- Life Application The Pentagon has a rotational symmetry of 72° When it is rotated at 72°, it maps onto itself. Save the Leaning Tower of Pisa! It is 3025 and the leaning tower of Pisa is now leaning too much. Now, the people decide that there is a need to reconstruct the tower. Find the angle you need to rotate the leaning tower to make the tower perpendicular to the ground by using a protractor. Translation By Andi Deng Vocab • Vector --A quantity having direction as well as magnitude – Initial point – starting point of the vector – Terminal point – ending point of the vector – Component form/ coordinate vector– horizontal and vertical values < a, b > • Coordinate notation– (x,y) (x+a, y+b) Andi’s explanation • It’s pretty much a thing move to another place. • See example: A house moved to a park from a city by a witch. ( not in scale) Vector component form/ coordinate vector: (15, 3) 3 15 Coordinate notation– (x,y) (x+15, y+3) Relating to the World y 8 • Mobile house 7 6 5 4 • A moving house 3 2 1 –8 • Another moving house –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 1 2 3 4 5 6 7 8 x Vector component form/ coordinate vector: (10, 5) practice Coordinate notation – (x,y) (x+10, y+5) • Make the house transfer 10 boxes to right and y 5 boxes up. 8 7 6 5 4 3 2 1 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 1 2 3 4 5 6 7 8 x Bibliography • • • • • • • • http://www.mathnstuff.com/gif/7x7not.gif http://2.bp.blogspot.com/AZAjgubNMhA/TwgsiY86q5I/AAAAAAAADSc/5vIUw095L7k/s1600/taj+mahel+%25285%25 29.jpg http://frank.itlab.us/france_2005/small_france/aug_02_north_bridge_2129.jpg http://lh5.ggpht.com/hFHVVeQ1ivFZv98SQyzvkeJcaaZJDHppDqa_zPdMYclne6eCsrMGaXHBNybBGRxzjqNSBPJZZzKNQzQHzLenos=s285-c http://1.bp.blogspot.com/G6CJXrFzR48/TgvUn_GsxCI/AAAAAAAAAxA/9C6zW9cOKZ8/s1600/Onassis+resevoir+reflectio ns+600.jpg http://uploads.static.vosizneias.com/2013/03/Rite-Aid.jpg http://brentwood.thefuntimesguide.com/images/blogs/new-brentwood-park-concordroad.jpg http://www.gbrionline.org/wp-content/uploads/2012/09/Dubai-Marina.jpg Bibliography • http://www.ysjournal.com/articles/2009/2/7/images/YoungScientistsJ_2009_ 2_7_35_57766_u13.jpg • http://math.slu.edu/prep08/images/prep08/thumb/3/38/Climatron.jpg/300p x-Climatron.jpg • http://cdn.indulgy.com/zB/wC/H7/VakilMosqueShabestanweb.jpg • http://1.bp.blogspot.com/hM6SOjPC0wg/UQzqZTCkPCI/AAAAAAAAALs/1rHEdIUX90M/s1600/YoungScie ntistsJ_2009_2_7_35_57766_u12.jpg • http://1howmany.com/files/uploads/images/pentagon-2.jpg • http://hdwallpaperfreedownload.com/wp-content/gallery/new-y