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Warm Up

Learn about applying similarity transformations in coordinate plane, determining figure similarity, and proving circle similarity with examples. Practice problems included.

aaronlewis
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Warm Up

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  1. Warm Up Find the image point when the indicated transformation is applied to the given pre-image point. x, 1. (x, y) → (x + 3, y – 1); (2, 4) 2. (x, y) → (x, –y); (–2, 1) 1 1 y 3. (x, y) → (3x, 3y); (–5, 0) 3 3 ; (3, -6) 4. (x, y) →

  2. Objectives Draw and describe similarity transformations in the coordinate plane. Use properties of similarity transformations to determine whether polygons are similar and to prove circles similar.

  3. A transformation that produces similar figures is a similarity transformation. Similarity transformation: a dilation or a composite of one or more dilations and one or more congruence transformations. Two figures are similar if and only if there is a similarity transformation that maps one figure to the other figure.

  4. Remember! Translations, reflections, and rotations are congruence transformations.

  5. Example 1: Drawing and Describing Dilations A. Apply the dilation D to the polygon with the given vertices. Describe the dilation. D: (x, y) → (3x, 3y)A(1, 1), B(3, 1), C(3, 2) dilation with center (0, 0) and scale factor 3

  6. Example 1: Continued B. Apply the dilation D to the polygon with the given vertices. Describe the dilation. x, D: (x, y) → 3 3 3 4 y 4 4 P(–8, 4), Q(–4, 8), R(4, 4) dilation with center (0, 0) and scale factor

  7. Example 2 : Determining Whether Polygons are Similar Determine whether the polygons with the given vertices are similar. A. A(–6, -6), B(-6, 3), C(3, 3), D(3, -6) and H(-2, -2), J(-2, 1), K(1, 1), L(1, -2)

  8. Example 2: Continued B. P(2, 0), Q(2, 4), R(4, 4), S(4, 0) and W(5, 0), X(5, 10), Y(8, 10), Z(8, 0).

  9. Example 2: Continued C. A(1, 2), B(2, 2), C(1, 4) and D(4, -6), E(6, -6), F(4, -2)

  10. Example 2: Continued D. F(3, 3), G(3, 6), H(9, 3), J(9, –3) and S(–1, 1), T(–1, 2), U(–3, 1), V(–3, –1).

  11. Check It Out! Example 2 Determine whether the polygons with the given vertices are similar : A(2, -1), B(3, -1), C(3, -4) and P(3, 6), Q(3, 9), R(12, 9).

  12. Example 3: Proving Circles Similar A. Circle A with center (0, 0) and radius 1 is similar to circle B with center (0, 6) and radius 3.

  13. Example 3: Continued Circle A can be mapped to circle A’ by a translation: (x, y) → (x, y + 6). Circle A’ and circle B both have center (0, 6). Then circle A’ can be mapped to circle B by a dilation with center (0, 6) and scale factor 3. So circle A and circle B are similar. B. Circle C with center (0, –3) and radius 2 is similar to circle D with center (5, 1) and radius 5.

  14. Example 3: Continued Circle C can be mapped to circle C’ by a translation: (x, y) → (x + 5, y + 4). Circle C’ and circle D both have center (5, 1).Then circle C’ can be mapped to circle D by a dilation with center (5, 1) and scale factor 2.5. So circle C and circle D are similar.

  15. Check It Out! Example 3 Prove that circle A with center (2, 1) and radius 4 is similar to circle B with center (-1, -1) and radius 2. Circle A can be mapped to circle A' by a translation: (x, y) → (x - 3, y - 2). Then circle A' can be mapped to circle B by a dilation with center (-1, -1) and scale factor So, circles A and B are similar. 1 2

  16. Lesson Quiz : Part-III 4. Prove that circle A with center (0, 4) and radius 4 is similar to circle B with center (-2, -7) and radius 6. Circle A can be mapped to circle A’ by a translation: (x, y) → (x - 2, y - 11). Circle A’ and circle B both have center (-2, -7). Then circle A’ can be mapped to circle B by a dilation with center (-2, -7) and scale factor So circle A and circle B are similar. 3 2

  17. Multiplying and Dividing Radicals • To Multiply: or: • To Divide: or: or:

  18. Simplifying Radicals: • Simplify • Simplify • Simplify Can’t be simplified because there is no perfect square when factored

  19. Rationalizing Radicals • We can NEVER leave a fraction with a radical in the denominator. • To rationalize we multiply the entire fraction by 1, which is the denominator/the denominator. This is the denominator This = 1

  20. Assignment Pg. 476 (3-12)

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