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Around the World with Transformations Table of Contents Rotations: Slides 3-13 (Jake Campbell) Translations: Slides 14-19 (Alison Sorkenn) Dilations: Slides 20-29 (Eric Park) Reflections: Slides 30-40 (Angela Hong) Tessellations: Slides 41-47 (Conor McCorry) Rotations Rotations ● ● ● ● A rotation is when a figure is turned about a fixed point Rotations are isometries (Segment and angle measures remain congruent) This point is called the center of rotation The amount of degrees that the figure is rotated about that point is the angle of rotation ● Unless noted, the angle of rotation is always counter-clockwise Rotations on the Coordinate Plane ● To rotate points counterclockwise around the origin when given coordinates, use the equations below ● 90°- (x,y) = (-y,x) ● 180°- (x,y) = (-x,-y) ● 270° - (x,y) = (y,-x) ● For clockwise rotations the equations are ● 90°- (x,y) = (y,-x) ● 180°- (x,y) = (-x,-y) ● 270° - (x,y) = (-y,x) Activity: Rotate ΔABC 180° about the origin A= (1,1) B= (4,1) C= (5,3) A′= (-1,-1) B′= (-4,-1) C′= (-5,-3) Now plot the points of the image and preimage. C A B’ C’ A’ B Finding Angles of Rotation ● Reflecting over two intersecting lines is the same as rotating ● To find the angle of rotation, find the angle of where the lines meet and multiply it by 2 In image on the left, what is the angle of rotation? 32° Answer: 64° Rotational Symmetry ● If a figure can be rotated onto itself in 180° or less, it has rotational symmetry ● All regular polygons have rotational symmetry ● You can find the angle of rotation for regular polygons by by using the formula 360/n (number of sides) ● To determine if an irregular shape has rotational symmetry, you have to eyeball it ● If the irregular shape has rotational symmetry, figure out how many ways the shape can map on to itself with rotations less than or equal to 360° ● The number of times a shape can map onto itself is known as the order ● To find the angle of rotational symmetry, use the equation 360/o (order) Real-Life Examples of Rotational Symmetry This Hindu (the primary religion of Bangladesh) swastika has 90° rotational symmetry The flag of Bangladesh has 180° rotational symmetry Does this flower, native to Bangladesh, have rotational symmetry? If so, what is the angle of rotation? (Assume all petals are congruent) Answer: The flower has 72° rotational symmetry. Since it can be rotated onto itself 5 times within a 360° rotation, its order is 5. When we use the equation 360/o, we get 72°. Rotating Around a Point Using a Ruler and Protractor (Original Artwork) 1. Rotate triangle ABC 90° clockwise around point D 2. Draw a line segment from point C to the point of rotation (P) 3. Using a protractor, create the angle of rotation (90°) using point P as the vertex. Make sure that the angle is in the direction that you’re rotating (clockwise) 5. Measure the length of CP 4. Finish drawing in the angle 6. Construct point C’ the length that you previously obtained along the angle that you drew in. 7. Erase angle CPC’ and repeat steps 1-6 for point A 8. Repeat steps 1-6 for point B. The final, rotated triangle should look like this. Translations Key Words to Know •When an object is translated, all points are moved the same distance and in the same direction. •Translation is a transformation that maps original points to their final points •Vector is a quantity that has both direction and magnitude •A translation is an isometry, which means that the lengths stay the same. <6,-2> 6 units right, 2 units down ● Reflecting an object twice over parallel lines results in a translation. Translating Objects •When an object is translated, it can be moved to any quadrant, as long as it stays in the same direction and is the y same size. x •Original objects are known as pre-images and translated objects are the images of the original. Rules to Use for Translations •There are three different ways to write out translations. •Coordinate Notation: (x,y) → (x+a,y+b) -represents how the coordinates are translated; adding/subtracting x and y •Vector in Component Form: <6,-2> -shows movement of the coordinates •Matrices: x y -rows and columns showing x and y coordinates -use matrix addition to find translated points Finding Image of Pre-Image •If you are given a pre-image with points on a graph and a rule, you will have to figure out the points of the image. •Ex: A(-8,7) B(-8,1) C(-1,7) D(-1,1) Pre-Image A Image A’ C’ B’ D’ C •Rule= (x,y) → (x+10,y+2) •Find the coordinates of the image and the vector that describes the new image. •The x-coordinate will increase by 10 units and the y-coordinate will increase by 2 units. ● Final Points: A(2,9) B(2,3) C(9,9) D(9,3) B D Word Problem •The Brazil soccer team is playing a game and needs to score a goal. One of the players with the ball sees an open player and decides to pass it to him. Write the vector in component form that describes the translating of the ball from player to player. Dilations Dilations Dilation- with a center of C and a scale factor of K, is a transformation that maps every point from the pre-image to their corresponding points of the dilated image 1. If p is not the center point C, then the image point P’ lies on CP. The scale factor k is a positive number such that k = CP’/CP, and K =/=1 2. If P is the center point C, then P = P’ Reduction- A dilation is a reduction if: 0 < K < 1 Enlargement- A dilation is an enlargement if: K > 1 Finding the scale factor of an image Put the value of the dilated image over the value of the preimage Dilated image/ Preimage 10/5 (Then reduce the fraction) 2/1 The scale factor of this dilation is 2 Preimage 5 inches Dilated image 10 inches Dilating Images with the origin as the center Matrix- rectangular arrangement of numbers in rows and columns Scalar multiplication- multiplying a matrix by a real number Multiply the x and y coordinates of the primage by the scale factor to get the coordinates of the dilated image Ex: (Scale factor 3) Coordinates of Preimage: (0,4) (5,-1) (-2,2) X [ 0 5 -2 ] Y [4 -1 2 ] [ (3)0 (3)5 (3)-2 ] [ (3)4 (3)-1 (3)2 ] Coordinates of Dilated image: (0,12)(15,-3)(-6,6) [ 0 15 -6] [ 12 -3 6] Dilating Images with the origin as the center Use scalar multiplication to find the coordinates of the dilated image Given: Scale factor = 2. (0,4) (2,3) (3,1) Answer: [ (2)0 (2)2 (2)3] [ (2)4 (2)3 (2)1] [ 0 4 6] (0,8) (4,6) (6,2) [ 8 6 2] Dilating Images with the origin as the center Plot the points on the graph from the previous slide: Dilating Images with the origin as the center Now you try! Dilate the coordinates (1,3) (2,4) (5,1). (First graph these points). With a scale factor of 3 and the center as the origin Answers: A’=(3,9) B’=(6,12) C’=(15,3) Dilating Images with the origin as the center The British soccer team Manchester United uses three people to pass the ball between. They each travel back the same distance as the others. If the original coordinates of the three people are: (1,4)(3,6)(4,2). And the coordinates of the three people after traveling back an equal amount: (4,16)(12,24)(16,8). Find the scale factor. Answer: Dilated coordinate over preimage coordinate: 4/1= 4 is the scale factor Challenge yourself Imagine you are a toy modeler. You are trying to make a replica of the building Big Ben. Big Ben is approximately 316 feet. The height of your model is 66 inches. The width of Big Ben is 50 feet. Find the width of your model. Answer: First convert feet to inches (You want your toy model in inches). (3792 inches, and 600 inches for width) Then put the dilated image over the preimage. 66/3792 = 0.017 which is your scale factor. Then multiply 50 by your scale factor. (which gets you is the answer 0.87inches) Which Dilating an image with a center that is not the origin When the center of the dilation is not the origin, you can’t simply multiply the x and y coordinates of the preimage by the scale factor. Use this equation when dilating an image on a coordinate plane. (This equation does work when the center is the origin, but that just adds extra work) K(x-a)+a K(y-b)+b Where: (x,y) are the coordinates of a point on a preimage (a,b) coordinates of the center of the dilation K is the scale factor Reflections ● A reflection is a transformation in which an image is reflected over a line ● The line of reflection is the line over which the image is reflected ● A figure has a line of symmetry if it can be mapped onto itself by a reflection over the line ○ Reflections are isometries, meaning that the lengths of the preimage (original) and image (transformed figure) are preserved An easier way to think about reflections is through the comparison of mirrors. ● The object you hold up to a mirror is the preimage, and its reflection in the mirror is the image ● The line of reflection acts as the mirror itself, because it is what reflects the image onto a new space. MATH is reflected over the line y = -x Reflections Above, the image reflects itself over the line x = 0 The Louvre Pyramid Zis reflected in the water at night Determining Lines of Symmetry A figure has a line of symmetry if one side of the shape can be reflected over the line and be congruent to the other. In other words, imagine “folding” the figure over the line, like a piece of paper. NO This is not a line of symmetry. YES This is a line of symmetry. Determining Lines of Symmetry Triangles can have 3, 1, or 0 lines of symmetry Equilateral Isosceles Right Hexagons can have different lines of symmetry depending on their shape Irregular Regular Irregular In a regular polygon, the number of lines of symmetry is equal to the number of sides the polygon has, because regular polygons have equal sides and angles. 3 Sides = 3 Lines 4 Sides = 4 Lines 5 Sides = 5 Lines 6 Sides = 6 Lines The Palace of Versailles is an example of a symmetrical landmark. The line of symmetry runs vertically through the middle of the palace and splits the figure into two congruent halves. The Eiffel Tower also has a vertical line of symmetry. Reflecting a Preimage With Coordinates When reflecting a preimage over a line on a coordinate plane, you can either… 1. Pick a vertex on the preimage, and count the number of units it takes to reach the line of reflection by using “rise/run”. Make sure you are counting box-by-box! 2. Count that same number of units from the line of reflection in the opposite direction of the preimage and plot the new point. 3. Repeat steps 1-2 for all the points of the preimage. 4. Connect the plotted points to form the new image. Use These Rules: OR Reflect over x-axis (x, y) (x, - y) Reflect over y-axis (x, y) (- x, y) Reflect over y = x (x, y) (y, x) Reflect over x = y (x, y) (- y, - x) Write An Equation for a Line of Reflection For lines… 1. Choose a point on the preimage and locate the corresponding point on the image 2. Use midpoint formula to find the midpoint of the two points 3. Find the perpendicular slope of the two points by taking the opposite recipricol of the regular slope 4. Use the perpendicular slope as “m” to create a slope-intercept equation by plugging in the midpoint coordinates as “x” and “y” For polygons… 1. Choose a point on the preimage and locate the corresponding point on the image 2. Use midpoint formula to find the midpoint of the two points 3. Repeat steps 1-2 4. Find the slope of the two plotted points and use it to create an equation by plugging in one of the midpoint coordinates as “x” and “y” Finding Minimum Distance Find point C on line m so that AC + BC is of minimum distance. A B m 1. Reflect one of the points over the line 2. Use distance formula or measure with a ruler to find the distance between the reflected and the other unreflected point (A’ and B, or A and B’) 3. The distance calculated is the minimum distance B A A’ B A’ Minimum Distance = 7 cm Now You Try! Jocelyn’s town is installing new water pipes on Chestnut Avenue. Her dad is in charge of the construction project, and asks her to help. The pipes must stretch from House A to House D, but they must also reach the pump of the reservoir on the other side of the street in order to draw water. What is the least amount of piping Jocelyn’s dad can use? A For this problem: Assume 1 cm = 10 yards B C RESERVOIR D Now You Try: Answer 1. The pipes must go from House A to House D. All other Houses in between are irrelevent to the problem. 2. Reflect one of the houses over the street, and use a ruler to measure the distance between House A’ and House D. Distance formula cannot be used in this particular problem because no coordinates were given. A D A D A’ 3. The distance measured is 5 cm, and since 1 cm = 10 yards, the least amount of piping Jocelyn’s dad must use is 50 yards. * Not to Scale * Minimum Distance = 5 cm 1 cm = 10 yards 5 cm = 50 yards Tessellations Classification of Frieze Pattern Key Vocabulary T A frieze pattern or border pattern is a pattern that extends to the left and right in such a way that the pattern can be mapped onto itself by a horizontal translation. Tessellations are isometries, because the lengths of the preimage and image are unchanged. Translation TR Translation 180 degree rotation TG Translation and horizontal glide reflection TV Translation and vertical line reflection THG Translation, 180 degree rotation, and horizontal glide reflection TRVG Translation, 180 degree rotation, vertical line reflection, and horizontal glide reflection TRHVG Translation, 180 degree rotation, horizontal line reflection, vertical line reflection, and horizontal line reflection Determining what Tessellates Using the equation 180(n-2)/n, you can determine what objects tessellate n=number of sides If the answer you get is a factor of 360, then the object will tessellate Determine if a dodecagon tessellates 180(12-2)/12 180(10)/12 1800/12 150 360/150=2.4 This is an example of a tesselation. Determine the Pattern TV 1. 2. T or THG 1. Name the Transformation that maps A onto B 2. Name the Transformation that maps B onto E A B E C F D 3. Name the Transformation that maps A onto E A1. T A2. TG A3. 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