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Understanding Dilations: Scale Factors, Symmetry, and Transformations in Geometry

This lesson focuses on geometric dilations, exploring scale factors, enlargement, and reduction. Students will learn how to identify lines of symmetry, understand angles of rotation, and apply transformation rules such as translations, reflections, and rotations. The class will also highlight practical applications, such as scaling models, including the iconic Willis Tower, and solving proportions. Engage with interactive worksheets and review essential concepts to prepare for upcoming tests.

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Understanding Dilations: Scale Factors, Symmetry, and Transformations in Geometry

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  1. 9.5 Dilations Brett Solberg AHS ‘11-’12

  2. Warm-up • Draw all the lines of symmetry. What is the angle of rotation? • 1 2 3

  3. Today’s Agenda • Warm-up/HW Review • Chapter 9 Review • 9.5 Dilations • Scale Factor • enlargement • reduction

  4. Announcements • Today • Review • 9.5 • Next Class • Test • Homework • Worksheet Packet • 9.2 Graphing Packet • Review Worksheet

  5. Transformations • Translations (slide) • rule of (4, -5) • Reflections (flip) • x-axis (x, y) → (x, -y) • y-axis (x, y) → (-x, y) • line

  6. Rotations • Center of Rotation • protractor • compass • Rotating about Origin • 90˚ clockwise (x, y) → (y, -x) • 90˚ counterclockwise (x, y) → (-y, x) • 180˚ either way (x, y) → (-x, -y)

  7. Rotation • Rotate 90˚, 180˚, 270˚

  8. Symmetry • Reflection Symmetry • lines of symmetry • Rotational Symmetry • angle of rotation

  9. Dilation • A dilation is a transformation whose image and preimage are similar.

  10. Dilation • Eyes Dilating • Scale Models

  11. Vocabulary • Scale Factor – determines the size change from the original figure to its image. • The dilation is an if the scale factor is greater than 1. • The dilation is a if the scale factor is between 0 and 1. enlargement reduction

  12. Example 1 • Find the scale factor. The red square is the dilation of the blue square. 4 4 2 2

  13. Example 3 • Scale Factor = 2

  14. Example 2

  15. Willis Tower • Built in 1974, originally called the Sears tower. • Stands at 1451ft (442m) • Was the world’s tallest building for 25 years • Currently the tallest building in USA.

  16. Willis Tower • The ratio of height to width is 6.45:1. • The building is 1451 ft tall. What is the width?

  17. Willis Tower LEGO Model (Sears Tower)scale model

  18. Willis Tower Scale Model

  19. Solving Proportions The ratio of height to width is 6.45:1 If the width of the model is 3 ft, and the model is similar to the actual tower, how tall is the model?

  20. Actual Dimensions • width 75 inches • length 189 inches • Model dimensions • width 7 inches • length 20 inches • Does a dilation occur? Jaguar Proportion

  21. Assignments • Assignments • Worksheet Packet (50 points) • 9.2 Graphing Packet (10 points) • Review Worksheet (10 Points) • Warm-ups (10 points) • Extra Credit Opportunities • pg523 #1 - 29 • Poster • translation • reflection • rotation • dilation

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