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Understanding Dilations in Geometry

cade-mcfadden avatar
cade-mcfadden
25 diapositives · 362 Views · Nov 08, 2014

Learn about dilations in geometry, which change the size but not the shape of a figure. Discover how to identify, draw, and work with dilations using clear examples and step-by-step instructions.

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Understanding Dilations in Geometry
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  1. Dilations 12-7 Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Are you ready? 1. Translate the triangle with vertices A(2, –1), B(4, 3), and C(–5, 4) along the vector <2, 2>. 2. ∆ABC ~ ∆JKL. Find the value of JK.

  3. Objective TSW identify and draw dilations.

  4. Vocabulary center of dilation enlargement reduction

  5. A dilation is a transformation that changes the size of a figure but not the shape. The image and the preimage of a figure under a dilation are similar.

  6. Example 1: Identifying Dilations Tell whether each transformation appears to be a dilation. Explain. A. B.

  7. Example 2 Tell whether each transformation appears to be a dilation. Explain. a. b.

  8. Helpful Hint For a dilation with scale factor k, if k > 0, the figure is not turned or flipped. If k < 0, the figure is rotated by 180°.

  9. A dilation enlarges or reduces all dimensions proportionally. A dilation with a scale factor greater than 1 is an enlargement, or expansion. A dilation with a scale factor greater than 0 but less than 1 is a reduction, or contraction.

  10. Example 3: Drawing Dilations Copy the figure and the center of dilation P. Draw the image of ∆WXYZ under a dilation with a scale factor of 2. Step 1 Draw a line through P and each vertex. Step 2 On each line, mark twice the distance from P to the vertex. Step 3 Connect the vertices of the image.

  11. Example 3: Drawing Dilations You should get something that looks like this: W’ X’ Y’ Z’

  12. Example 4 Copy the figure and the center of dilation. Draw the dilation of RSTU using center Q and a scale factor of 3. Step 1 Draw a line through Q and each vertex. Step 2 On each line, mark twice the distance from Q to the vertex. Step 3 Connect the vertices of the image.

  13. Example 4 R’ S’ T’ U’

  14. Example 5: Drawing Dilations On a sketch of a flower, 4 in. represent 1 in. on the actual flower. If the flower has a 3 in. diameter in the sketch, find the diameter of the actual flower.

  15. Example 6: Art Application An artist is creating a large painting from a photograph by dividing the photograph into squares and dilating each square by a scale factor of 4. If the photograph is 20 cm by 25 cm, what is the perimeter of the painting?

  16. Example 7 What if…? An artist is creating a large painting from a photograph into square and dilating each square by a factor of 4. Suppose the photograph is a square with sides of length 10 in. Find the area of the painting.

  17. If the scale factor of a dilation is negative, the preimage is rotated by 180°. For k > 0, a dilation with a scale factor of –k is equivalent to the composition of a dilation with a scale factor of k that is rotated 180° about the center of dilation.

  18. Example 8: Drawing Dilations in the Coordinate Plane Draw the image of the triangle with vertices P(–4, 4), Q(–2, –2), and R(4, 0) under a dilation with a scale factor of centered at the origin.

  19. Q’ R’ P’ Example 8 Continued It should look like this: P R Q

  20. Example 9 Draw the image of the triangle with vertices R(0, 0), S(4, 0), T(2, -2), and U(–2, –2) under a dilation centered at the origin with a scale factor of .

  21. T’ U’ S’ R’ S R T U Example 9 Continued It should look like this

  22. Check Your Understanding 1. Tell whether the transformation appears to be a dilation.

  23. Lesson Quiz: Part II 2. A rectangle on a transparency has length 6cm and width 4 cm and with 4 cm. On the transparency 1 cm represents 12 cm on the projection. Find the perimeter of the rectangle in the projection. 3. Draw the image of the triangle with vertices E(2, 1), F(1, 2), and G(–2, 2) under a dilation with a scale factor of –2 centered at the origin.