Lesson 12-13: Properties of Similarity Transformations
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1 Lesson 12-13: Properties of Similarity Transformations Learning Target I can define a similarity transformation as the composition of basic rigid motions and dilations. I can define two figures to be similar if there is a similarity transformation that takes one to the other. Opening Activity The diagram below shows a dilation of the plane or doesn t it? Explain your answer. Definition A similarity transformation (or similarity) is a composition of a number of and/or. The of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1. Characteristics Similar figures should look the same, but one is a,,, or relative to the other, corresponding angles are equal in measure Definition 2 Two figures in a plane are similar if there exists a taking one figure onto the other figure. Examples Similar Non-Examples
2 Example 1. Given triangle ABC as shown on the diagram of the coordinate plane. a) State the coordinates of all vertices of ABC. A B C b) Perform a translation so that vertex A maps to the origin. State this transformation in T a,b notation. State all new coordinates. A B C c) Next, dilate the image A B C from the origin using a scale factor of 1 3. State this with D P,r notation. State all new coordinates. A B C d) Without measuring, what is the ratio of A B : AB? e) Without measuring what can we say about angles ABC and A B C. What happens to angle measure after a similarity transformation? f) Finally, translate the image A B C so that the vertex A maps to the original point A. State the translation with T a,b notation. State all new coordinates. A B C g) Using transformations, describe how the resulting image A B C relates to the original figure ABC.
3 Example 2. Given the coordinate plane shown, identify a similarity transformation, if one exists, mapping X onto Y. Lesson Summary Two figures are similar if there exists a similarity transformation that maps one figure onto the other. A similarity transformation is a composition of a finite number of dilations or rigid motions. Properties of similarity transformations: There is a scale factor r for G, so that for any pair of points P and Q with images P = G(P) and Q = G(Q), then P Q = rpq. In other words, THE LENGTH OF A SIDE OF THE PRE-IMAGE IS MULTIPLIED BY THE SCALE FACTOR. A similarity transformation sends angles to angles of equal measure. In other words, CORRESPONDING ANGLES OF SIMILAR FIGURES ARE EQUAL.
4 Lesson 12-13: Properties of Similarity Transformations Classwork Exercise 1. A similarity transformation for triangle STU is described by r. Locate and label the image of STU under the similarity transformation D O, 1 r 2 y=x( STU) where the center of dilation is the origin. Original Coordinates ry = x D0, 1/2 S(, ) S (, ) S (, ) T(, ) T (, ) T (, ) U(, ) U (, ) U (, )
5 Exercise 2. Given O(0,0) and quadrilateral BCDE, with B( 5,1), C( 6, 1), D( 4, 1), and E( 4,2), what are the coordinates of the vertices of the image of BCDE under the similarity transformation r x axis (D O,3 (BCDE))? Original Coordinates D O,3 (BCDE) r x axis B(, ) B (, ) B (, ) C(, ) C (, ) C (, ) D(, ) D (, ) D (, ) E(, ) E (, ) E (, ) Exercise 3. Given the coordinate plane shown, identify a similarity transformation, if one exists, that maps ABCD onto A B C D. If one does not exist, explain why.
6 Exercise 4. A similarity transformation consists of a reflection over line l, followed by a dilation from O with a scale factor of r = 3 and result in G H I. 4 a) Find GH, IH, and IG in centimeters. Without measuring, calculate G H, I H and I G. b) m GIH = (4x + 19), m GHI = (3x 12), and m H G I = (x + 1). Find x and m G I H. Exercise 5. Similarity transformation G consists of a reflection across line l, followed by a dilation centered at P with scale factor r = 3. l If the original triangle had a perimeter of 16 units, what would the perimeter of the image be?
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