4Translation 1) PP’ = QQ’ 2) PP’ || QQ’, or PP’ and QQ’ are collinear. DefinitionPREIMAGEIt’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.IMAGEExamplesPP’1) PP’ = QQ’2) PP’ || QQ’, or PP’ and QQ’ are collinear.QQ’
5Double reflectionsIt’s when you reflect an image twice, then you’ll get a image that is transformed.reflectingtranslation
6Component form and vector Definition of component form:ExampleThe component form of a vector combines the horizontal and vertical components.VECTORDefinition of vector:COMPONENT FORMA vector is a quantity that has both direction and magnitude or size.COMPONENT FORM
7How 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.KThen count how much from j to k it moves from the x and y axis. Then you have your component form.43ANSWER:COMPONENT FORM OF JK IS <-6, 4>2J1-6-5-4-3-2-1
8PROBLEMS!Find the component form of vector AB.BA<-9, 5>
9Equation to translation The equation for translation is:(x,y) (x+a, x+b)a = horizontal changeB = vertical changeSo basically what your doing is adding the component form to a coordinate pair.
10Example:Find the new vertices after the transformation of parallelogram ABCD. The component form is <2, -4>.ADA (6,7); B (7,2); C (10,2); D (9,7) A’ – (6 + 2, ) = (8, 3) B’ – (7 + 2, ) = (9, -2) C’ – (10 + 2, ) = (12, -2) D’ – (9 + 2, ) = (11, 3)BC
11Practice problem! Answers: Component form: <3, 2> A’ = (3, 9) 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!AAnswers:Component form: <3, 2>A’ = (3, 9)B’ = (4, 6)C’ = (7, 8)CB
12MatricesAnother 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>…
13Example! E’ = (3, 6) F’ = (7, 4) G’ = (8, 7) Use matrices to find the translation of figure EFG.Use vector FG for the component form.GE’ = (3, 6)F’ = (7, 4)G’ = (8, 7)EF
14PRACTICE! Answers: Component form: <-1, 6> A’ = (5, 13) Using matrices, find the coordinates after a translation, using the vector CD.ADAnswers:Component form: <-1, 6>A’ = (5, 13)B’ = (4, 7)C’ = (7, 7)D’ = (6, 13)BC
15A word problem Component form: <-1, 5> 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.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)BCAD
16Translations in the Rainforest 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!
18A 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 rotationAngle of rotation- rays drawn from the center of rotation to a pointRotational symmetry- a figure in a plane that can be mapped onto itself by a clockwise rotation of 180° or lessRotationsHere is an example of a rotation.The center of rotation is point P
19One 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
20Another way to rotateFirst, 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 verticesUse 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 2Lastly, repeat the steps for each vertex and connect the dots to form the final figure.
21This diagram illustrates the steps to find one of the point on the final figure.
22In 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.
23Theorem 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
24Sample Problems Which of these figures rotational symmetry? What is the angle of rotation?Answer: The recycling logoAnswer: 100 degrees
25Rotations in the Rainforest When the parrot changes course it rotates around his right wing. This wing acts as the point of rotation. The parrot rotates 90° clockwise.
26Rotation Activity! Rotate the sloth 180° clockwise on the tree branch It should look something like this:
27Let’s learn about Reflections 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 congruentLet’s learn about Reflections
28When reflecting a point over the x-axis, y-axis, y=x, or y=-x just follow these formulas!
29How 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)
30Reflect 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.7654321(-1,6)(-4,2)(-1,2)
31How many lines of symmetry do these shapes have? 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 verticesHow many lines of symmetry do these shapes have?Regular Pentagon has 5 lines of symmetryRectangle has 2 lines of symmetryIsosceles Trapezoid has 1 line of symmetry
32Reflections in the Rainforest 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.
33Using your knowledge of reflections and lines of symmetry, decorate your own butterfly!
34DilationsA 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!
35This would be a reduction, since the afterimage is smaller than the pre-image.
36When the center is the origin, you can find the coordinates of the afterimage by multiplying the coordinates of the pre-image by the scale factorThe 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)
37You 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
38Now you try!Answer:This is an enlargement because the image is larger than the pre-image.The scale factor is 3
39Dilations in the Rainforest 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!
40What is a Tessellation?A tessellation is a repeating pattern of figures that completely covers a plane without any overlaps or gapsYou can make a tessellation only by using isometrics (Translation, rotations, and reflections)Triangles and Quadrilaterals can always tessellate
41What 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.
42Tessellation VocabVector- 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 imageRegular Polygon- a polygon that is equiangular and equilateralSemi 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:
43Try to Tessellate! Can you tessellate this caterpillar? Your tessellation should look like this:
44Tessellations 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!