Transformations Geometry

Transcription

1 Transformations Geometry Preimage the original figure in the transformation of a figure in a plane. Image the new figure that results from the transformation of a figure in a plane. Example: If function h : x 2x 3, find the Image of 8 and the Preimage of 11. Solution: What would the outcome be What value of x would give if x 8? an outcome of 11? h : , h : x 2x 3 11 The image of 8 is 13. 2x 14 x 7. The preimage of 11 is 7. Isometry a transformation that preserves length. Mapping an operation that matches each element of a set with another element, its image, in the same set. Transformation the operation that maps, or moves, a preimage onto an image. Three basic transformations are reflections, rotations, and translations. The three main Transformations are: Reflection Flip! Rotation Turn! Translation Slide!

2 Reflection type of transformation that uses a line that acts like a mirror, called a line of reflection, with a preimage reflected over the line to form a new image. {Flip} A reflection is a FLIP over a line. Every point is the same distance from the central line! The reflection has the same size as the original image. Line of reflection the mirror line. A reflection is an isometry. Example: Given rectangle ABCD, write the coordinates of each of the points by reflection in: 7 6 D C A B a) The x-axis A 1, 1, B 5, 1, C 5, 6, D 1, 6 As the rectangle is reflected over the x -axis: A ' 1, 1, B ' 5, 1, C ' 5, 6, D ' 1, 6. b) The y-axis A 1, 1, B 5, 1, C 5, 6, D 1, 6 As the rectangle is reflected over the y -axis: A' 1, 1, B ' 5, 1, C ' 5, 6, D ' 1, 6. c) The line y = x. A 1, 1, B 5, 1, C 5, 6, D 1, 6 As the rectangle is reflected over the y=x: A ' 1, 1, B ' 1, 5, C ' 6, 5, D ' 6, 1.

3 Notice the patterns: reflection over the x-axis change the sign of the y- coordinate, (x, y) (x, -y) reflection over the y-axis change the sign of the x-coordinate, (x, y) (-x, y) reflection over y = x switch the order of the x and y-coordinates. (x, y) (y, x) And if all else fails, just fold your sheet of paper along the mirror line and hold it up to the light! Line of symmetry a line that a figure in the plane has if the figure can be mapped onto itself by a reflection in the line. Examples: Determine and draw all lines of symmetry in the following figures. Example: Reflect over the y-axis: Solution:

4 Example: Notice that some letters possess vertical line symmetry, some possess horizontal line symmetry, and some possess BOTH vertical and horizontal line symmetry. A point reflection exists when a figure is built around a single point called the center of the figure. It is a direct isometry. P x, y x, y. Rotation a type of transformation in which a figure is turned about a fixed point, called a center of rotation. {Turn} Center of rotation the fixed point. Angle of rotation the angle formed when rays are drawn from the center of rotation to a point and its image. Counterclockwise rotation is considered positive and clockwise is considered negative. "Rotation" means turning around a center. The distance from the center to any point on the shape stays the same. Every point makes a circle around the center. A rotation is an isometry.

5 Examples: O is the center of regular pentagon ABCDE. State the images of A, B, C, D, and E under each rotation. a) R O,144 b) R O,-72 A Rotate counterclockwise Rotate clockwise 144 about point O. 72 about point O. E 72 O B A C B D A E B A C E C B D C D A D C E B. E D. Examples: a) Each of these figures has rotation symmetry. Estimate the center of rotation and the angle of rotation? a) For each shape, the center of rotation is the center of the figure. The angles of rotation, from left to right, are 120, 180, 120, and 90. b) Do the regular polygons have rotation symmetry? For each polygon, what are the center and angle of rotation? b) For each shape, the center of rotation is the center of the figure. The angles of rotation, from left to right, are 120, 180, 120, and 90. c) Name the vertices of the image of KLM after a rotation of 90. K(4, 2), L(1, 3), and M(2, 1). c) K (-2, 4), L (-3, 1), and M (-1, 2).

6 Notice the patterns: rotation of 90 change the sign of the y-coordinate then switch the x and y-coordinates, (x, y) (-y, x) rotation of 180 change the signs of the x and y- coordinates, (x, y) (-x, -y) rotation of 270 change the sign of the x-coordinate then switch the x and y-coordinates. (x, y) (y, -x) Translation a type of transformation that maps every two points P and Q in the plane to points P and Q, so that the following two properties are true. 1. PP = QQ. 2. PP ' QQ ' or PP ' and QQ ' are collinear. {Slide} When you are sliding down a water slide, you are experiencing a translation. Your body is moving a given distance (the length of the slide) in a given direction. You do not change your size, shape or the direction in which you are facing. "Translation" simply means Moving without rotating, resizing or anything else, just moving. Theorem: A translation is an isometry. Every point of the shape must move: the same distance, in the same direction. Examples: Explain the meaning of: a) (x, y) (x 7, y + 3) Slide the original point(s) by moving it left 7 units and up 3 units. {mapping notation} b) T (-7, 3) or T -7, 3 (x, y) Slide the original point(s) by moving it left 7 units and up 3 units. {symbol notation}

7 Examples: a) Which of the following lettered figures are translations of the shape of the purple arrow? Chose ALL that apply. Solution: a, c, and e only. b) Which of the following translations best describes the diagram at the left? i. 3 units right and 2 units down. ii. 3 units left and 2 units up. iii. 3 units left and 2 units down. Solution: (i) 3 right and 2 down. Glide reflections a transformation in which every point P is mapped onto a point P by the following two steps. 1. A translation maps P to P. 2. A reflection in a line k parallel to the direction of the translation maps P to P. When a translation (a slide or glide) and a reflection are performed one after the other, a transformation called a glide reflection is produced. In a glide reflection, the line of reflection is parallel to the direction of the translation. It does not matter whether you glide first and then reflect, or reflect first and then glide. This transformation is commutative. Since translations and reflections are both isometries, a glide reflection is also an isometry.

8 Examples: a) Does this paw print illustration depict a glide reflection? Solution: Yes. b) Examine the graph below. Is triangle A"B"C" a glide reflection of triangle ABC? Solution: Yes. c) A triangle has vertices A(3,2), B(4,1) and C(4,3). What are the coordinates of point B under a glide reflection, T 0,1 r x? Solution: After the reflection, B (-4, 1). After the translation, B (-4, 2). d) Given triangle ABC: A(1,4), B(3,7), C(5,1). Graph and label the following composition: T 5, 2 r x axis.

9 Line Reflection Point Reflection Translations Rotations Dilations Opposite isometry Properties preserved: distance angle measure parallelism collinearity midpoint Reverse Orientation (letter order changed) Direct isometry Direct isometry Direct isometry Properties preserved: distance angle measure parallelism collinearity midpoint Same Orientation (letter order the same) Properties preserved: distance angle measure parallelism collinearity midpoint Same Orientation (letter order the same) Properties preserved: distance angle measure parallelism collinearity midpoint Same Orientation (letter order the same) NOT an isometry. Figures are similar. Properties preserved: angle measure parallelism collinearity midpoint Lengths not same. Same Orientation (letter order the same) See additional notes on notations under Geometry notes on teacher s MAC webpage. Glide Reflection Opposite isometry Properties preserved: distance angle measure parallelism collinearity midpoint Reverse Orientation (letter order changed)

10 Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry) is easy to recognize, because one half is the reflection of the other half. With Rotational Symmetry, the shape or image can be rotated and it still looks the same. The image is rotated (around a central point) so that it appears 2 or more times. The number of times it appears is called its order. Point Symmetry is when every part has a matching part: the same distance from the central point but in the opposite direction. Point symmetry exists when a figure is built around a single point called the center of the figure. For every point in the figure, there is another point found directly opposite it on the other side of the center. Note: it s the same as "Rotational Symmetry of Order 2". A simple test to determine whether a figure has point symmetry is to turn it upside-down and see if it looks the same. A figure that has point symmetry is unchanged in appearance by a 180 rotation. Examples: Line symmetry: The white line down the center is the Line of Symmetry.

11 a) Which of the following lettered items possesses line symmetry? Consider vertical and/or horizontal symmetry. Chose ALL that apply. a. the letter F Solution: b, c, d, and e. b. an equilateral triangle c. the letter X d. the word BOO e. a square f. the word WALLY g. the letter Q b) If the left wings of this symmetric butterfly have a total of 32 beige scallops around their edges, how many total beige scallops does the butterfly possess? Solution: 64 Examples: Rotational symmetry: The US Bronze Star Medal A Dartboard has Rotational has Rotational Symmetry of Order 5. Symmetry of Order 10. a) Does this sign have rotational symmetry? If so, what are the degrees of symmetry? Solution: Yes, 120 and 240.

12 b) Which of the following letters or objects possesses rotational symmetry? Chose ALL that apply. a. letter F Solution: b, c, d, and e. b. an equilateral triangle c. letter X d. a rectangle e. a square f. letter W g. the letter Q Examples: Point symmetry: HRotational Symmetry of Order a) A sign by a swimming hole displays the message shown at the left. The message is saying that there is no swimming on Mondays. What is special about the way the message is written? Solution: It has point symmetry. It reads the same when turned upside-down. b) Which of the following lettered items possesses point symmetry? Choae ALL that apply. a. the letter D Solution: b, c, d, and g. b. a square c. the letter S d. the word e. the letter B f. the word DAD g. the letter Z

13 c) Which of these cards has point symmetry? Solution: The 2 of spades. A dilation used to create an image larger than the original is called an enlargement. A dilation used to create an image smaller than the original is called a reduction. A figure and its dilation are similar figures. Reduction a dilation with 0 k 1. Enlargement a dilation with k 1. A dilation of scalar factor k whose center of dilation is the origin may be written: Dk (x, y) = (kx, ky). Examples: a) Draw the dilation image of triangle ABC with the center of dilation at the origin and a scale factor of 2. OBSERVE: Notice how EVERY coordinate of the original triangle has been multiplied by the scale factor (x2). HINT: Dilations involve multiplication!

14 b) Draw the dilation image of pentagon ABCDE with the center of dilation at the origin and a scale factor of 1/3. OBSERVE: Notice how EVERY coordinate of the original pentagon has been multiplied by the scale factor (1/3). HINT: Multiplying by 1/3 is the same as dividing by 3! c) Draw the dilation image of rectangle EFGH with the center of dilation at point E and a scale factor of 1/2. For this example, the center of the dilation is NOT the origin. The center of dilation is a vertex of the original figure. OBSERVE: Point E and its image are the same. It is important to observe the distance from the center of the dilation, E, to the other points of the figure. Notice EF = 6 and E'F' = 3. HINT: Be sure to measure distances for this problem.