Rotations A rotation is an isometry where shapes rotate around a fixed point in a circular motion, whether clockwise or counter clockwise
Angle of Rotation : rays drawn from the center of rotation to a point and its image form an angle -If center of rotation is origin: R 90° (x,y) = (-y,x) R 180° (x,y) = (-x,-y) R 270° (x,y) = (y,-x) R -90° (x,y) = (y,-x) Center of Rotation : the fixed point of a rotation PPoint P is the center of rotation Rotational Symmetry: when a figure can be mapped onto itself by a clockwise rotation of 180 degrees or less -An equilateral triangle can be mapped onto itself by 120 degrees Rotations Vocabulary
Center of Rotation: (0,0) Angle of Rotation: 270° Vertices: A (-4,4,)A’ (4,4) B (-2,4)B’ (4,2) C (-2,2)C’ (2,2) D (-4,2)D’ (2,4) A B C D A’ B’ C’ D’D’
B A C ED F Center of Rotation: Point F Angle of Rotation: 80 degrees
Tweety wants to go into his cage. Rotate Tweety 110 degrees about point P, (9,4), so that he is in his cage. Find A’, B’, C’, D’, and E’. Help Tweety Vertices: A (16.5, 7) B (14, 6.5) C (12.5, 7) D (13, 9) E (14.5, 9.5)
Real Life Application The top of Tweety’s bird cage has rotational symmetry. It can be mapped onto itself at 36 degrees.
Bibliography "Looney Tunes." SAT 400. N.p., n.d. Web. 23 Apr. 2013.. "EK Success Wavy Circle Large Punch." BGPayne Crafts. N.p., n.d. Web. 23 Apr. 2013..
Vocabulary Reflection – an image over a line, that almost acts like a mirror. Line of Reflection – the which acts like a mirror in a reflection. Line of symmetry – a figure that can be mapped onto itself by a reflection in the line. Isometry – transformation which the two figures are congruent.
Line of Symmetry 3 Sides 3 Lines 4 Sides 2 Lines 4 Sides 4 Lines 3 Sides 1 Line In a regular polygon, the number of lines of symmetry is equal to the number of side. In a non regular polygon, one must just count.
Minimum Distance To find the minimum distance, you reflect one of the initial points (point A), then you connect A’ to point B. where that line crosses the x-axis will be the minimum distance point C. A B C A’ A (-1,5) B (5,1) A’ (-1,-5) C (4,0) What is the equation of the line A’B?
Real Life Application If a character from the Simpsons were to look into a mirror they would see their face reflected back at them.
GSP Activity Go onto GSP and make sure you have graph up. Then plot A(-1,-2) and B (8,-4). Next find the minimum distance and the equation of A’B. Do the same for A (1,4), B (8,3).
Tom and Jerry’s Translations By: Eric Spiniello
Words to Know: Translation: a type of transformation where every point of a pre image is moved a certain distance is a certain direction to form an image. The image is congruent to the pre image, it is just moved. Initial Point: The starting point of a vector. Terminal Point: The end point of a vector Vector: a quantity that has both magnitude and direction. Component Form: is made up by the horizontal and vertical components of a vector. For example, the rule (x,y)>(x+2,y-3) becomes in component form. Coordinate notation: tells you the distance on the x and y axis you should move each point. For example, (x,y)>>>(x+a,y+b). “A” represents the amount of units you moved on the x axis and “B” represents the amount that you moved on the y axis.
Mammy Two Shoes Mathematical Examples In this example, each point slides 7 units left and 3 down. This means that the rule is (x,y)>>(x-7,y-3). In component/vector form it would be and in coordinate notation (x,y)>>(x-7,y-3).
Mammy Two Shoes Mathematical Examples You can also find this by using matrices. First, you must take the coordinates of A B C and D and record them in a matrix. The x coordinate plots go on the top, with the y on the bottom. [A B C D] [A B C D] [A B C D] [2 4 5 2] + [-7 -7 -7 -7] = [-5 -3 -2 -5] [4 4 2 1] + [-3 -3 -3 -3] = [ 1 1 -1 -2] Since the rule is (x,y)>>(x-7,y-3), we added -7 to the x coordinates and -3 to the y coordinates. This tells us that the new coordinates for the image are A’= (-5,1) B’= (-3,1) C’= (-2,-1) D’= (-5,-2)
Tom’s Translation Concepts If you are given a pre image at point (3,-2) and a rule (x,y)>>(x+5,y-2) then you would start at point (3,-2) and count 5 units to the right and 2 units down on a coordinate plane. So the coordinates of the image would be (8,-4). If you are given the image at point (8,-4) and a rule (x,y)>>(x+5,y-2), then you would subtract 5 from 8 and add 2 to -4. This would make the coordinates of the pre image (3,-2).
Jerry’s GSP Activity Under graph, click “show grid”. Create any shape of your choice. Label the points. Highlight your shape. Go to the “Transform” window and select “Translate”. On the new window select “Rectangular” under “Translation Vector”. Write in 7cm for the horizontal and 2cm for the vertical fixed distances. This will be the number of units your new image will translate. Click “Translate”. Your new image is a translation from the original pre image. It should look like the example.
Jerry’s GSP Activity Question 1.What is the rule for the translation you just made? 2.How would you write that rule in component form?
Real World Applications In cartoons, translations are everywhere. For example, as Tom runs after Jerry, both characters are translating and moving across the screen.
Bibliography tomjerrynew.blogspot.com (Title Photo) tomjerrynew.blogspot.com http://atminhd.com/tom-and-jerry-wallpaper-download-hd.html (Tom and Jerry Second Slide) http://atminhd.com/tom-and-jerry-wallpaper-download-hd.html http://www.regentsprep.org/Regents/math/geometry/GT2/Trans.htm (Translation Diagram) http://www.regentsprep.org/Regents/math/geometry/GT2/Trans.htm http://en.wikipedia.org/wiki/Mammy_Two_Shoes (Mammy Two Shoes) http://en.wikipedia.org/wiki/Mammy_Two_Shoes protagonist.wikia.com (Tom) protagonist.wikia.com mugen.wikia.com (Jerry) mugen.wikia.com http://www.goldenagecartoons.com/reviews/2008/tjtales4/ (Tom and Jerry Confused) http://www.goldenagecartoons.com/reviews/2008/tjtales4/ www.toptimelinecovers.com (Tom chasing Jerry) www.toptimelinecovers.com http://www.regentsprep.org/Regents/math/geometry/GT2/Trans.htm (Math Information) http://www.regentsprep.org/Regents/math/geometry/GT2/Trans.htm Geometry Textbook (Math Information)
What's a Tesserration Raggy? A tessellation is the process of creating a two-dimensional plane using the repetition of a geometric shape with no overlaps and no gaps Example: Around any vertex or corner point in a tessellation the measure of all angles must equal 360 degrees In this case four 90 degree angles
Jeepers Gang, look at that Tessellation ! There are many tessellations that appear everywhere The most common being floor tilling The tessellation made in this floor tilling is made of regular squares alternating from white to black This is made of rectangular bricks on a wall on a street or of a house This tessellation is know as a 4.4.4 tessellation because of the amount of shapes and their # of sides 44 4
How to Create a Scooby Snack (Tessellation) 1.Find a picture that you want to tessellate like this one. A square, rectangle, right triangle, or regular triangle would be the easiest. 2.Start by placing it into a new Photoshop document 1.The size should be U.S. paper size which is selected when you select File New then in the Present drop down menu select U.S. Paper
3. To get the picture in the document copy the picture form any online source and then in the Edit drop down menu on the top left select paste and the picture should be right there. 4. After this press the Crtl key simultaneously with the T key and the picture should be able to be resized. Hold shift and on the corners adjust it to your proffered size. After this press Crtl and the D key 5. Move the picture with the tool by pressing v and move it to the top left leaving room in between the top and the left edge of the paper 6. Then ¾ down the right hand side there is a tab called layers select that 7. Right click the layer that says Layer 1 and select duplicate
8. The new image should appear right on top of the other one, just take this image and move it to be right next to the other one to the right 9. Keep doing this until you have enough to make a row across the paper but not touching the right edge 10. Then select on the layers tab the top most image, right click and select merge down until all you have is the one layer and the background 11. Then duplicate this layer comprised of all the copied pictures and duplicate that. Then place this copy underneath the other image
12. Keep doing this until the whole page is filled. The final project should look somewhat like this.
Terms To Know Dilation – Transformation that produces a shape that is different in size Scale Factor – Ratio of corresponding sides of an image over a pre- image (K) Reduction – If the scale factor is less than one Enlargement – If the scale factor is greater than one
Terms to know Cont. Center of Dilation – A fixed point where all points are dilated Equation: Dk(x,y)=(kx,ky) K = OP’/OP
Properties preserved Angle measures remain the same (Parallelism) Parallel lines remain parallel (Colinearity) Points stay on the same lines (Midpoint) Midpoints remain the same in each figure (Orientation) Lettering order remains the same
Example of a Reduction Scale Factor 1/2 Multiplying with Matrices ABCABC X =573 Y =633 1/2 A’B’C’ X =184.108.40.206 Y =31.51.5
Example of a Enlargement Scale Factor 2 Enlargement Multiplying with Matrices ABCABC X =573 Y =633 2 A’B’C’ X =10146 Y =1266
Real Life Application It has been discovered that a full grown Box Jellyfish can be 300 cm long and 25 cm wide. It has also been found that a baby Box Jellyfish can measure 15 cm long. Find the scale Factor of the length Using the Scale factor find the width of the baby jellyfish Is this a reduction or an enlargement 300 cm 15 cm
GSP Activity First Click ‘graph’ and then click on ‘form grid’ Next Create a triangle Label the triangle ABC Measure the lengths of the triangle Highlight the triangle and click ‘transform’ then ‘dilate’ Use a scale factor of ½ Make this triangle A’B’C’ Measure the Lengths of the new triangle Move the triangle around examining what happens
GSP Questions Is the new triangle a reduction or a enlargement? What happens when you move triangle ABC? If you dilate triangle A’B’C’ with the same scale factor what happens? What Happens when you move the original triangle at the end of the steps above?
Answers Dilation Answers Real Life applications answer – 300/15 = 20/1 1.25 cm Enlargement GSP Answers When you move triangle ABC triangle A’B’C’ should move making the side lengths ½ triangle ABC Reduction The new triangle will have half the side lengths of triangle A’B’C’ A’B’C’ should be half the side length of Triangle ABC and The new triangle should be half the side lengths triangle A’B’C’
Answers Translation Answers 1: (x,y)>>>>(x+7,y+2) 2: Reflection Answers What is the equation of the line? y=x-4 A(-1,-2) and B (8,-4), C (2,0). y= -2/9x+4/9 A (1,4), B (8,3), C (5,0). y= -1/7x+5/7