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Understanding Similarity Transformations and Dilations

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Learn about similarity transformations, including translations, rotations, reflections, and dilations. Understand the concept of dilations, center of dilation, and scale factor. Practice drawing dilations and calculating scale factors.

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Understanding Similarity Transformations and Dilations

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  1. No Clickers Bellwork • Find the geometric mean of 5 & 18 • Find the geometric mean of 3 & 44 • Solve for x • Solve for x, if AB & CD are parallel • What point is twice as far from the origin as (3, 5)? 5 3 x x-2 2x-1 C A x 8 B 12 D

  2. Bellwork Solution • Find the geometric mean of 5 & 18

  3. Bellwork Solution • Find the geometric mean of 3 & 44

  4. Bellwork Solution • Solve for x 5 3 x x-2

  5. Bellwork Solution C • Solve for x, if AB & CD are parallel A x 2x-1 8 12 B D

  6. Bellwork Solution • What point is twice as far from the origin as (3, 5)?

  7. Perform Similiarity Transformations Section 6.7

  8. Test on Wednesday

  9. The Concept • We’ve covered most of chapter 6, but we have yet to apply our understanding of similarity to objects on the coordinate plane. • Today we’re going to use our understanding of similarity and transformations to discuss similarity transformations.

  10. Review • We’ve seen three kinds of transformations thus far • Translations • Shifts or moves up or down and right or left • Rotations • Object rotations a direction and angle about the origin • Reflections • Flips of an object either over the x-axis or the y-axis • The last one that we will learn about is dilations

  11. Definitions • Dilation • Special kind of transformation that stretches or shrinks a figure to create a similar figure • Figures are either reduced or enlarged • Type of similarity transformation

  12. Definitions • Center of Dilation • Fixed point with which the object is dilated • Scale factor of dilation • Ratio of a side length of the image to the corresponding side length of the original figure

  13. Coordinate Notation • We prefer to be able to notate for dilations • For dilations centered at the origin • (x,y)(kx,ky), where k is a scale factor • If 0<k<1, reduction • If k>1, enlargement

  14. Drawing a Dilation • Draw a dilation of an object with vertices (0,2), (5, 3) & (5,-3) using a scale factor of 2

  15. Drawing a Dilation • Draw a dilation of an object with vertices (4,6), (2, 4) & (6,-6) using a scale factor of 1/2

  16. Example Draw a dilation of a quadrilateral ABCD with vertices A(2,2), B(4,2), C(4,0), D(0,-2). Use a scale factor of 1.5 and label the object FGHJ

  17. Scale or k factor • We’ve discussed scale factor before and defined it as • The quotient of a side length of the second object and the corresponding side length of the first object • This property of dilations is no different • For example, find the scale factor of these two objects 2 1

  18. Example • We can also determine k factor from points • Find the k factor between these two objects What do we need in order to give an accurate answer

  19. Example • Is the green object a dilation of the yellow one? How do we know?

  20. Practical Example • You are using a photo quality printer to enlarge a digital picture. The picture on the computer screen is 6 centimeters by 6 centimeters. The printed image is 15 cm by 15 cm. What is the scale factor of the enlargement?

  21. Homework • 6.7 Exercises • 1, 2-8 even, 9-23, 25, 26

  22. Example Draw a dilation of a quadrilateral ABCD with vertices A(-3,5), B(3,4), C(4,-2), D(-3,-2). Use a scale factor of 1.5 and label the object FGHJ

  23. Most Important Points • Definition of Dilation • Bounds for the k scalar • Performing Dilations • Finding k factor from points

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