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Understanding Similarity Transformations: Dilations Explained

Learn about dilations in similarity transformations, including reductions, enlargements, scale factor, and center of dilation. Discover how to determine missing sides and use proportions in similar figures.

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Understanding Similarity Transformations: Dilations Explained

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  1. Chapter 6: Similarity Perform Similarity Transformations 6.7

  2. Transformations • Remember previously we talked about 3 types of CONGRUENCE transformations, in other words, the transformations performed created congruent figures • Rotation, Reflection, and translation • Now we will also discussed Dilations

  3. – Perform Similarity Transformations • a dilation is a transformation in which a figure and its image are similar • the ratio of the new image to the original figure is called the scale factor • a dilation with a scale factor greater than 1 is called an enlargement • a dilation with a scale factor less than 1 is called a reduction • a figure is enlarged or reduced with respect to a fixed point called the center of dilation.

  4. Enlargements • When you have a photograph enlarged, you make a similar photograph. X 3

  5. Reductions • A photograph can also be shrunk to produce a slide. 4

  6. Determine the length of the unknown side. 15 12 ? 4 3 9

  7. These triangles differ by a factor of 3. 15 3= 5 15 12 ? 4 3 9

  8. DILATIONS • REDUCTION VS. ENLARGEMENT • For k, a scale factor we write (x, y)  (kx, ky) • If 0 < k < 1 the dilation is a reduction • If k > 1 the dilation is an enlargement

  9. Yesterday you constructed an enlargement similar to this • Let ABCD have A(2, 1), B(4, 1), C(4, -1), D(1, -1) • Scale factor: 2 • You used the GM to show that they were similar.

  10. Center of Dilation (Usually the origin) • To show that two objects are similar, we draw a line from the origin to the object further away, if the line passes through the closer object in the similar vertex then the objects are similar.

  11. When changing the size of a figure, will the angles of the figure also change? ? 40 70 ? ? 70

  12. Nope! Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees.If the size of the angles were increased,the sum would exceed 180 degrees. 40 40 70 70 70 70

  13. We can verify this fact by placing the smaller triangle inside the larger triangle. 40 40 70 70 70 70

  14. The 40 degree angles are congruent. 40 70 70 70 70

  15. The 70 degree angles are congruent. 40 40 70 70 70 70 70

  16. The other 70 degree angles are congruent. 4 40 70 70 70 70 70

  17. Find the length of the missing side. 50 ? 30 6 40 8

  18. This looks messy. Let’s translate the two triangles. 50 ? 30 6 40 8

  19. Now “things” are easier to see. 50 30 ? 6 40 8

  20. The common factor between these triangles is 5. 50 30 ? 6 40 8

  21. So the length of the missing side is…?

  22. That’s right! It’s ten! 50 30 10 6 40 8

  23. WARM-UP R U A GEN!US? 1)  Take your shoe size (no half sizes, round up). 2)  Multiply your number by 5. 3)  Add 50. 4)  Multiply by 20. 5)  Add 1013 to that number. 6)  Subtract the year you were born (ex: 1996). 7)  Let me guess what you got!!

  24. Similarity is used to answer real life questions. • Suppose that you wanted to find the height of this tree.

  25. Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.

  26. You can measure the length of the tree’s shadow. 10 feet

  27. Then, measure the length of your shadow. 10 feet 2 feet

  28. If you know how tall you are, then you can determine how tall the tree is. 6 ft 10 feet 2 feet

  29. The tree must be 30 ft tall. Boy, that’s a tall tree! 6 ft 10 feet 2 feet

  30. Similar figures “work” just like equivalent fractions. 30 5 11 66

  31. These numerators and denominators differ by a factor of 6. 30 6 5 6 11 66

  32. Two equivalent fractions are called a proportion. 30 5 11 66

  33. Similar Figures • So, similar figures are two figures that are the same shape and whose sides are proportional.

  34. (Page 371 on Blue Text)

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