9-1 SIMILAR POLYGONS Objectives: Identify similar polygons and use ratios to solve problems involving similar polygons. Similar Polygons Two polygons are similar polygons if corresponding  s are congruent and if the lengths of corresponding sides are proportional. Two polygons are similar polygons if corresponding  s are congruent and if the lengths of corresponding sides are proportional. Diagram Definition Symbols

9-1 SIMILAR POLYGONS Objectives: Identify similar polygons and use ratios to solve problems involving similar polygons. Proportions Cross Products Property

9-1 SIMILAR POLYGONS Objective: Use ratios to solve problems involving similar polygons.  b. What is the extended proportion for the ratios of the lengths of corresponding sides?

9-1 SIMILAR POLYGONS Objective: Use ratios to solve problems involving similar polygons. Are the polygons similar? If they are, write a similarity statement and give the scale factor. The figures are not similar. 

9-1 SIMILAR POLYGONS Objective: Use ratios to solve problems involving similar polygons.   Your class is making a rectangular poster for a rally. The poster’s design is 6 in. wide by 10 in. high. What are the dimensions of the largest complete poster that will fit in a space 3-ft wide by 4-ft high? 28.8 inches wide by 48 inches high

9-1 SIMILAR POLYGONS Objective: Use ratios to solve problems involving similar polygons. a. The diagram shows a scale drawing of the Golden Gate Bridge in San Francisco. What is the actual height of the towers above the roadway? b. The Space Needle in Seattle is 605 ft tall. A classmate wants to make a scale drawing of the Space Needle on an 8 ½ in.–by-11 in. sheet of paper. He decides to use the scale 1 in. = 50 ft. Is this a reasonable scale? Explain. Use 0.8 cm as the height of the towers.  No, using a scale of 1 in. = 50 ft., the paper must be more than 12 in. tall. No, using a scale of 1 in. = 50 ft., the paper must be more than 12 in. tall. 

9-1 SIMILAR POLYGONS JT P M N Objective: Use ratios to solve problems involving similar polygons.

9-1 SIMILAR POLYGONS Objective: Use ratios to solve problems involving similar polygons. 3. The scale drawing at the right is part of a floor plan for a home. The scale is 1 cm = 2 ft. What are the actual dimensions of the dining room and kitchen? Do you know HOW? 9 cm 3.75 cm

9-1 SIMILAR POLYGONS Objective: Use ratios to solve problems involving similar polygons. 4. Vocabulary What does the scale on a scale drawing indicate? Do you understand? 5. Explain Mathematical Ideas (1)(G) The polygons shown are similar. Two friends write the similarity statements shown. Which friend is not correct? Explain. A. TRUV  NPQU B. RUVT  QUNP The scale indicates how many units of length of the actual object are represented by each unit of length in the drawing.

9-1 SIMILAR POLYGONS Objective: Use ratios to solve problems involving similar polygons. 6. Justify Mathematical Arguments (1)(G) Is similarity reflexive? Transitive? Symmetric? Justify your reasoning. Do you understand? Every figure is similar to itself, so similarity is reflexive. If figure 1  figure 2 and figure 2  figure 3, then figure 1  figure 3, so similarity is transitive. If figure 1  figure 2 and figure 2  figure 3, then figure 1  figure 3, so similarity is transitive. If figure 1  figure 2, then figure 2  figure 1, so similarity is symmetric.

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity. Your friend says that she performed a composition of transformations to map  ABC to  A’B’C’. Describe the composition of transformations. Translate  ABC until points B and B’ coincide. Then dilate by the appropriate scale factor until the two figures overlap. Translate  ABC until points B and B’ coincide. Then dilate by the appropriate scale factor until the two figures overlap.

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity. Composing a Rigid Transformation and a Dilation  LMN has vertices L(-4, 2), M(-3, -3), and N(-1, 1). Suppose the triangle is translated right 4 units and up 2 units, and then dilated by a scale factor of 0.5 with center of dilation at the origin. Sketch the resulting image.

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity.

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity. If there is a dilation that maps one triangle to another, then they are similar. Determine whether the two triangles are similar. Explain your reasoning. a.  NSQ with vertices N(0, 3), S(3, 2), and Q(2, 1)  FRM with vertices F(2, 6), R(6, 4), and M(4, 2) b.  JLT with vertices J(-1, 3), L(-3, 2), and T (-1, 1)  MIH with vertices M(-2, 6), I(-6, 4), and H(-2, 2) Not similar; although a dilation with scale factor of 2 maps vertices S to R and Q to M, it does not map N to F. Not similar; although a dilation with scale factor of 2 maps vertices S to R and Q to M, it does not map N to F. Similar; a dilation with scale factor of 2 maps  JLT to  MIH. Similar; a dilation with scale factor of 2 maps  JLT to  MIH.

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity. Do you know HOW? 2. The length and width of a 5.7 in.-by-3.8 in. rectangular photo are each increased by x in. Can a similarity transformation map one of the images onto the other? Explain why or why not.

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity. 3. Identify whether the triangles are similar by determining if there is a similarity transformation that maps one to the other. If the triangles are similar, compare the ratio of their perimeters to the scale factor. a.  PQR with vertices P(3, 0), Q(-4, -2), and R(5, -1)  STV with vertices S(2.5, 0.5), T(1.5, 0), and V(-2, -1) The figures are not similar. Solve for x: Solve for lengths of sides: Solve for ratio of lengths of sides: Solve for perimeters of  s: Solve for ratio of perimeters:

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity. Do you understand? 4. Vocabulary Describe how the word dilation is used in areas outside of mathematics. How do these real-world applications relate to the mathematical definition? 5. Analyze Mathematical Relationships (1)(F) Your classmate claims that the order of the rigid transformation and the dilation that map  RST to  JKL does not matter. Do you agree? Explain. The pupils of your eyes dilate when you go from dark to bright locations or from bright to dark. The pupils are reduced or enlarged proportionally to form similar pupils.

9-2 SIMILARITY TRANSFORMATIONS Objective: To identify similarity transformations and verify properties of similarity. 6. Create Representations to Communicate Mathematical Ideas (1)(E) For TUV at the right, give the vertices of a similar triangle after a similarity transformation that uses at least one rigid motion. Do you understand? (Show your answer to your classmate or teacher for discussion and checking.)