Geometry unit 7.2
Download as PPT, PDF1 like1,110 views
This document is about similarity and similar polygons from a Holt Geometry textbook. It defines similar polygons as polygons with congruent angles and proportional side lengths. Examples are provided to identify similar polygons and write similarity statements. The similarity ratio of two similar polygons is defined as the ratio of corresponding side lengths. Applications include using proportions to find the length of a model based on its similarity to an actual object.
Geometry unit 7.2
- 1. Holt Geometry UUNNIITT 77..22 SSIIMMIILLAARRIITTYY
- 2. Warm Up 1. If ΔQRS @ ΔZYX, identify the pairs of congruent angles and the pairs of congruent sides. ÐQ @ ÐZ; ÐR @ ÐY; ÐS @ ÐX; QR @ ZY; RS @ YX; QS @ ZX Solve each proportion. 2. 3. x = 9 x = 18
- 3. Objectives Identify similar polygons. Apply properties of similar polygons to solve problems.
- 4. Vocabulary similar similar polygons similarity ratio
- 5. Figures that are similar (~) have the same shape but not necessarily the same size.
- 6. Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional.
- 7. Example 1: Describing Similar Polygons Identify the pairs of congruent angles and corresponding sides. ÐN @ ÐQ and ÐP @ ÐR. By the Third Angles Theorem, ÐM @ ÐT. 0.5
- 8. Check It Out! Example 1 Identify the pairs of congruent angles and corresponding sides. ÐB @ ÐG and ÐC @ ÐH. By the Third Angles Theorem, ÐA @ ÐJ.
- 9. A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons. The similarity ratio of ΔABC to ΔDEF is , or . The similarity ratio of ΔDEF to ΔABC is , or 2.
- 10. Writing Math Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order.
- 11. Example 2A: Identifying Similar Polygons Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. rectangles ABCD and EFGH
- 12. Example 2A Continued Step 1 Identify pairs of congruent angles. All Ðs of a rect. are rt. Ðs and are @. ÐA @ ÐE, ÐB @ ÐF, ÐC @ ÐG, and ÐD @ ÐH. Step 2 Compare corresponding sides. Thus the similarity ratio is , and rect. ABCD ~ rect. EFGH.
- 13. Example 2B: Identifying Similar Polygons Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. ΔABCD and ΔEFGH
- 14. Example 2B Continued Step 1 Identify pairs of congruent angles. ÐP @ ÐR and ÐS @ ÐW isos. Δ Step 2 Compare corresponding angles. mÐW = mÐS = 62° mÐT = 180° – 2(62°) = 56° Since no pairs of angles are congruent, the triangles are not similar.
- 15. Check It Out! Example 2 Determine if ΔJLM ~ ΔNPS. If so, write the similarity ratio and a similarity statement. Step 1 Identify pairs of congruent angles. ÐN @ ÐM, ÐL @ ÐP, ÐS @ ÐJ
- 16. Check It Out! Example 2 Continued Step 2 Compare corresponding sides. Thus the similarity ratio is , and ΔLMJ ~ ΔPNS.
- 17. Helpful Hint When you work with proportions, be sure the ratios compare corresponding measures.
- 18. Example 3: Hobby Application Find the length of the model to the nearest tenth of a centimeter. Let x be the length of the model in centimeters. The rectangular model of the racing car is similar to the rectangular racing car, so the corresponding lengths are proportional.
- 19. Example 3 Continued 5(6.3) = x(1.8) Cross Products Prop. 31.5 = 1.8x Simplify. 17.5 = x Divide both sides by 1.8. The length of the model is 17.5 centimeters.
- 20. Check It Out! Example 3 A boxcar has the dimensions shown. A model of the boxcar is 1.25 in. wide. Find the length of the model to the nearest inch.
- 21. Check It Out! Example 3 Continued 1.25(36.25) = x(9) Cross Products Prop. 45.3 = 9x Simplify. 5 » x Divide both sides by 9. The length of the model is approximately 5 inches.
- 22. Lesson Quiz: Part I 1. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. no 2. The ratio of a model sailboat’s dimensions to the actual boat’s dimensions is . If the length of the model is 10 inches, what is the length of the actual sailboat in feet? 25 ft
- 23. Lesson Quiz: Part II 3. Tell whether the following statement is sometimes, always, or never true. Two equilateral triangles are similar. Always
- 24. All rights belong to their respective owners. Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.