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Understanding Dilations: Concepts, Examples, and Applications

Explore dilations in geometry, from identifying to drawing them, with examples and practical applications. Learn about scale factors, similar figures, and the effects of dilations on shapes.

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Understanding Dilations: Concepts, Examples, and Applications

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  1. Dilations 9-7 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

  2. Warm Up 1. Translate the triangle with vertices A(2, –1), B(4, 3), and C(–5, 4) along the vector <2, 2>. A'(4,1), B'(6, 5),C(–3, 6) 2. ∆ABC ~ ∆JKL. Find the value of JK.

  3. Objective Identify and draw dilations.

  4. Vocabulary center of dilation enlargement reduction

  5. Recall that 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. No; the figures are not similar. Yes; the figures are similar and the image is not turned or flipped.

  7. Check It Out! Example 1 Tell whether each transformation appears to be a dilation. Explain. a. b. Yes, the figures are similar and the image is not turned or flipped. No, the figures are not similar.

  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 2: 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. W’ X’ Step 3 Connect the vertices of the image. Y’ Z’

  11. Check It Out! Example 2 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. R’ S’ Step 2 On each line, mark twice the distance from Q to the vertex. Step 3 Connect the vertices of the image. T’ U’

  12. Example 3: 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. The scale factor in the dilation is 4, so a 1 in. by 1 in. square of the actual flower is represented by a 4 in. by 4 in. square on the sketch. Let the actual diameter of the flower be d in. 3 = 4d d = 0.75 in.

  13. Check It Out! Example 3 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. The scale factor of the dilation is 4, so a 10 in. by 10 in. square on the photograph represents a 40 in. by 40 in. square on the painting. Find the area of the painting. A = l  w = 4(10)  4(10) = 40  40 = 1600 in2

  14. 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.

  15. The dilation of (x, y) is Example 4: 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.

  16. Q’ R’ P’ Example 4 Continued Graph the preimage and image. P R Q

  17. The dilation of (x, y) is Check It Out! Example 4 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 .

  18. T’ U’ S’ R’ S R T U Check It Out! Example 4 Continued Graph the preimage and image.

  19. 2. Copy ∆RST and the center of dilation. Draw the image of ∆RST under a dilation with a scale of . Lesson Quiz: Part I 1. Tell whether the transformation appears to be a dilation. yes

  20. Lesson Quiz: Part II 3. 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. 240 cm 4. 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.

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