Proportions and Similarity • Lessons 6-1, 6-2, and 6-3 Identify similar Key Vocabulary polygons, and use ratios and proportions to solve • proportion (p. 283) problems. • cross products (p. 283) • Lessons 6-4 and 6-5 Recognize and use • similar polygons (p. 289) proportional parts, corresponding perimeters, • scale factor (p. 290) altitudes, angle bisectors, and medians of similar • midsegment (p. 308) triangles to solve problems. • Lesson 6-6 Identify the characteristics of fractals and nongeometric iteration. Similar figures are used to represent various real-world situations involving a scale factor for the corresponding parts. For example, photography uses similar triangles to calculate distances from the lens to the object and to the image size. You will use similar triangles to solve problems about photography in Lesson 6-5. 280 Chapter 6 Proportions and Similarity Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 6. For Lesson 6-1, 6-3, and 6-4 Solve Rational Equations Solve each equation. (For review, see pages 737 and 738.) Ϫ y ϩ 2 2y 1. ᎏ2ᎏy Ϫ 4 ϭ 6 2. ᎏ5ᎏ ϭ ᎏx ᎏ4 3. ᎏ4ᎏ ϭ ᎏᎏ 4. ᎏᎏ ϭ ᎏ3ᎏ2 3 6 12 3 y Ϫ 1 4 y For Lesson 6-2 Slopes of Lines Find the slope of the line given the coordinates of two points on the line. (For review, see Lesson 3-3.) 5. (3, 5) and (0, Ϫ1) 6. (Ϫ6, Ϫ3) and (2, Ϫ3) 7. (Ϫ3, 4) and (2, Ϫ2) For Lesson 6-5 Show Lines Parallel Given the following information, determine whether a b. a State the postulate or theorem that justifies your answer. b (For review, see Lesson 3-5.) 1 234 c 8. Є1 Х Є8 56 78 9. Є3 Х Є6 10. Є5 Х Є3 For Lesson 6-6 Evaluate Expressions Evaluate each expression for n ϭ 1, 2, 3, and 4. (For review, see page 736.) 11. 2n 12. n2 Ϫ 2 13. 3n Ϫ 2 Proportions and Similarity Make this Foldable to record information about proportions and similarity in this chapter. Begin with one sheet of 11-inch ϫ 17-inch paper. Punch and Fold Divide Fold lengthwise. Open the flap and Leave space to draw lines to punch holes so it can divide the inside be placed in your into six equal parts. binder. Label 6-1 6-2 Label each part Proportions Put the name of 6-3 6-4 and using the lesson Similarity the chapter on the numbers. 6-5 6-6 front flap. Reading and Writing As you read and study the chapter, use the Foldable to write down questions you have about the concepts in each lesson. Leave room to record the answers to your questions. Chapter 6 Proportions and Similarity 281 Proportions • Write ratios. • Use properties of proportions. Vocabulary do artists use ratios? • ratio Stained-glass artist Louis Comfort Tiffany • proportion used geometric shapes in his designs. In • cross products a portion of Clematis Skylight shown at • extremes the right, rectangular shapes are used as • means the background for the flowers and vines. Tiffany also used ratio and proportion in the design of this piece. WRITE RATIOS A ratio is a comparison of two quantities. The ratio of a to b can be expressed ᎏaᎏ, where b is not zero. This ratio can also be written as a:b. b Study Tip Example 1 Write a Ratio Unit Ratios SOCCER The U.S. Census Bureau surveyed 8218 schools nationally about Ratios should be written their girls’ soccer programs. They found that 270,273 girls participated in a high in simplest form. A ratio school soccer program in the 1999–2000 school year. Find the ratio of girl soccer in which the denominator players per school to the nearest tenth. is 1 is called a unit ratio. To find this ratio, divide the number of girl athletes by the number of schools. number of girl soccer players ᎏᎏᎏᎏ ϭ ᎏ270,2ᎏ73 or about 32.9 number of schools 8,218 32.9 can be written as ᎏ32ᎏ.9. So, the ratio for this survey was 32.9 girl soccer players 1 for each school in the survey. Extended ratios can be used to compare three or more numbers. The expression a:b:c means that the ratio of the first two numbers is a:b, the ratio of the last two numbers is b:c, and the ratio of the first and last numbers is a:c. Standardized Example 2 Extended Ratios in Triangles Test Practice Multiple-Choice Test Item In a triangle, the ratio of the measures of three sides is 4:6:9, and its perimeter is 190 inches. Find the length of the longest side of the triangle. A 10 in.B 60 in.C 90 in.D 100 in. Read the Test Item You are asked to apply the ratio to the three sides of the triangle and the perimeter to find the longest side. 282 Chapter 6 Proportions and Similarity Christie’s Images Solve the Test Item Recall that equivalent fractions can be found by multiplying the numerator and the denominator by the same number. So, 2 : 3 ϭ ᎏ2ᎏ и ᎏxᎏ or ᎏ2ᎏx. Thus, we can rewrite Test-Taking Tip 3 x 3x Extended ratio problems 4:6:9 as 4x :6x :9x and use those measures for the sides of the triangle. Write an require a variable to be equation to represent the perimeter of the triangle as the sum of the measures of the common factor among its sides. the terms of the ratio. This 4x ϩ 6x ϩ 9x ϭ 190 Perimeter will enable you to write 9x an equation to solve the 19x ϭ 190 Combine like terms. 4x problem. ϭ x 10 Divide each side by 19. 6x Use this value of x to find the measures of the sides of the triangle. 4x ϭ 4(10) or 40 inches 6x ϭ 6(10) or 60 inches 9x ϭ 9(10) or 90 inches The longest side is 90 inches. The answer is C. CHECK Add the lengths of the sides to make sure that the perimeter is 190. 40 ϩ 60 ϩ 90 ϭ 190 ߛ Study Tip USE PROPERTIES OF PROPORTIONS An equation stating that two ratios Reading are equal is called a proportion. Equivalent fractions set equal to each other form a proportion. Since ᎏ2ᎏ and ᎏ6ᎏ are equivalent fractions, ᎏ2ᎏ ϭ ᎏ6ᎏ is a proportion. Mathematics 3 9 3 9 When a proportion is Every proportion has two cross products. The cross products in ᎏ2ᎏ ϭ ᎏ6ᎏ are 2 times written using colons, it is 3 9 read using the word to for 9 and 3 times 6. The extremes of the proportion are 2 and 9. The means are 3 and 6. the colon. For example, 2:3 is read 2 to 3. The means are the inside extremes ᎏ2ᎏ ϭ ᎏ6ᎏ means numbers, and the 3 9 extremes are the outside cross product of extremes 2(9) ϭ 3(6) cross product of means numbers. 18 ϭ 18 extremes Ά 2:3 ϭ 6:9 The product of the means equals the product of the extremes, so the cross products Ά means are equal. Consider the general case. a c ᎏᎏ ϭ ᎏᎏ b 0, d 0 b d a c (bd)ᎏᎏ ϭ (bd)ᎏᎏ Multiply each side by the common denominator, bd. b d da ϭ bc Simplify. ad ϭ bc Commutative Property Property of Proportions •Words For any numbers a and c and any nonzero numbers b and d, ᎏaᎏ ϭ ᎏcᎏ if and only if ad ϭ bc. b d • Examples ᎏ4ᎏ ϭ ᎏ1ᎏ2 if and only if 4 и 15 ϭ 5 и 12. 5 15 To solve a proportion means to find the value of the variable that makes the proportion true. www.geometryonline.com/extra_examples Lesson 6-1 Proportions 283 Example 3 Solve Proportions by Using Cross Products Solve each proportion. Ϫ Ϫ a. ᎏ3ᎏ ϭ ᎏxᎏ b. ᎏ3x ᎏ5 ϭ ᎏ13ᎏ 5 75 4 2 3 x 3x Ϫ 5 Ϫ13 ᎏᎏ ϭ ᎏᎏ Original proportion ᎏᎏ ϭ ᎏᎏ Original proportion 5 75 4 2 3(75) ϭ 5x Cross products (3x Ϫ 5)2 ϭ 4(Ϫ13) Cross products 225 ϭ 5x Multiply. 6x Ϫ 10 ϭϪ52 Simplify. 45 ϭ x Divide each side by 5. 6x ϭϪ42 Add 10 to each side. x ϭϪ7 Divide each side by 6. Proportions can be used to solve problems involving two objects that are said to be in proportion. This means that if you write ratios comparing the measures of all parts of one object with the measures of comparable parts of the other object, a true proportion would always exist. Study Tip Example 4 Solve Problems Using Proportions Common AVIATION Atwinjet airplane has a length of 78 meters and a wingspan of Misconception 90 meters. A toy model is made in proportion to the real airplane. If the The proportion shown in wingspan of the toy is 36 centimeters, find the length of the toy. Example 4 is not the only correct proportion. There Because the toy airplane and the real plane are in proportion, you can write a are four equivalent proportion to show the relationship between their measures. Since both ratios proportions: compare meters to centimeters, you need not convert all the lengths to the same ᎏaᎏ ϭ ᎏcᎏ, ᎏaᎏ ϭ ᎏcᎏ, unit of measure.