Similarity and Transformations Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry

Warm Up Find the image point when the indicated transformation is applied to the given pre-image point. 1. (x, y) → (x + 3, y – 1); (2, 4) (5, 3) 2. (x, y) → (x, –y); (–2, 1) (-2,-1) 3. (x, y) → (3x, 3y); (–5, 0) (-15,-0) 3 1 x, y 4. (x, y) → ; (3, -6) (1,-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.

Vocabulary similarity transformation

A transformation that produces similar figures is a similarity transformation. A similarity transformation is 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.

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

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

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

Check It Out! Example 1 Apply the dilation D : (x, y)→ polygon with vertices D(-8, 0), E(-8, -4), and F(-4, -8). Name the coordinates of the image points. Describe the dilation. to the 4 1 x, y D'(-2, 0), E'(-2, -1), F'(-1, -2); dilation with center (0, 0) and scale factor 1 4

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) Yes; ABCD maps to HJKL by a dilation: (x, y) → 1 3 x y ,

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). No; (x, y) → (2.5x, 2.5y) maps P to W, but not S to Z.

Example 2: Continued C. A(1, 2), B(2, 2), C(1, 4) and D(4, -6), E(6, -6), F(4, -2) Yes; ABC maps to A’B’C’ by a translation: (x, y) → (x + 1, y - 5). Then A’B’C’ maps to DEF by a dilation: (x, y) → (2x, 2y).

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). Yes; FGHJ maps to F’G’H’J’ by a reflection : (x, y) → (-x, y). Then F’G’H’J’ maps to STUV by a dilation: (x, y) 1 3 x y ,

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). The triangles are similar because ABC can be mapped to A'B'C' by a rotation: (x, y) → (-y, x), and then A'B'C' can be mapped to PQR by a dilation: (x, y) → (3x, 3y).

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.

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.

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.

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

Example 4: Application Eric wants to make a drawing to show how he will arrange three rugs in his room. The large rug is twice the size of each small rug. Eric will first draw the small rug in the lower left corner and then the large rug in the middle. How can he draw those rugs?

Example 4: Continue Place the lower left rug in the first quadrant of a coordinate plane in a convenient position. Apply the dilation with center (0, 0) and scale factor 2: (x, y) → (2x, 2y).

Check It Out! Example 4 What if…? How could Tia draw the middle flag to make it 4 times the size of each of the other flags? Apply the dilation with center (0, 0) and scale factor 4: (x, y) → (4x, 4y).

Lesson Quiz : Part-I 1. Apply the dilation D: (x, y) to the polygon with vertices A(2, 4), B(2, 6), and C(6, 4). Name the coordinates of the image points. Describe the dilation. A’(3, 6), B’(3, 9), C’(9, 6); dilation with center (0, 0) and scale factor 3 2

Lesson Quiz : Part-II Determine whether the polygons with the given vertices are similar. 2. A(-4, 4), B(6, 4), C(6, -4), D(-4, -4) and P(-2, 2), Q(4, 2), R(4, -2), S(-2, -2) No; (x, y) → (0.5x, 0.5y) maps A to P, but not B to Q. 3. A(2, 2), B(2, 4), C(6, 4) and D(3, -3), E(3, -6), F(9, -6) Yes; △ ABC maps to △ A’B’C’ by a reflection: (x, y) → (x, -y). Then △ A’B’C’ maps to △DEF by a dilation:(x, y) → (1.5x, 1.5y).

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 3 2 So circle A and circle B are similar.