Similarity and Transformations 7-2 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.

Congruence transformations are different from similarity Translations, reflections, and rotations are congruence transformations. Remember! Congruence transformations are different from similarity transformations because…. Angles and sides are exactly the same measurements when dealing with congruent triangles In similar triangles the angles are exactly the same measurements The sides are PROPORTIONAL (ratio/fractions must be simplified) Dilation is the ONLY transformation that is SIMILAR polygons

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 (enlargement) 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  

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 Reduction

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 , Reduction

On your Own : 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 Enlargement

On your Own : Part-I 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).