advertisement

Ch 6.1 • Ratio- comparison of two quantities – Ex: 3 4 or 3:4 • Ratios are UNITLESS – Means without units ( or units cancel out) • A ratio in which the denominator is 1 is called a unit ratio Proportion • Proportion – an equation stating that two ratios are equal – Equivalent fractions set equal to each other form proportions – Ex: 2 = 6 3 9 (equivalent fractions are fractions that reduce to the same quantity) Solve The U.S. Census Bureau surveyed 8,218 schools nationally about their girls’ soccer program. They found that 270,273 girls participated in a high school soccer program in the 1999-2000 school year. Find the ratio of girl soccer players per school to the nearest tenth. • Girls = 270,273 = 32.8879 = 32.9 schools 8,218 Solve In a triangle, the ratio of the measures of three sides is 4:6:9, and its perimeter is 190 inches. Find the longest side of the triangle. 4x + 6x + 9x = 190 19x = 190 X = 10 4x = 40 6x = 60 9x = 90 Pg 285 5) A replica of The Thinker is 10 inches tall. A statue of The Thinker, located in front of Grawemeyer Hall on the Belnap Campus of the University of Louisville in Kentucky, is 10 ft tall. What is the ratio of the replica to the statue in Louisville? 10 in. replica covert inches to ft with ratio 1ft 10 ft statue 12 in * 1 foot has 12 inches 10in * 1ft = 10ft 12in 12 1:12 1/12 10ft 12 . = 10ft * 1 = 1/12 10ft 12 10 ft The replica is 1/12 the size of the statue Cross Products • Every proportion has cross products – Ex: 2 = 6 3 9 Extremes (2) (9) = Means (3) (6) 18 = 18 The product of the means equals the product of the extremes, so the cross products are equal. Solve 1. 3 = x 5 75 2. 3x -5 = -13 4 2 Pg 286 YOU SOLVE 38. The ratios of the lengths of the sides of three polygons are given below. Make a conjecture about identifying each type of polygon. a. 2:2:3 b. 3:3:3:3 c. 4:5:4:5 40. In a golden rectangle, the ratio of the length of the rectangle to its width is approximately 1.618:1. Suppose a golden rectangle has a length of 12 centimeters. What is its width to the nearest tenth? Pg 286 YOU SOLVE 38. The ratios of the lengths of the sides of three polygons are given below. Make a conjecture about identifying each type of polygon. a. 2:2:3 b. 3:3:3:3 c. 4:5:4:5 40. In a golden rectangle, the ratio of the length of the rectangle to its width is approximately 1.618:1. Suppose a golden rectangle has a length of 12 centimeters. What is its width to the nearest tenth? Figures that are similar (~) have the same shape but not necessarily the same size. Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional. Example 1: Describing Similar Polygons Identify the pairs of congruent angles and corresponding sides. #1 N Q and P R. By the Third Angles Theorem, M T. Therefore: Tri NPM ~ Tri QRT 0.5 Identify the pairs of congruent angles and corresponding sides. #2 B G and C H. By the Third Angles Theorem, A J. A similarity ratio (scale factor) is the ratio of the lengths of the corresponding sides of two similar polygons. The similarity ratio of ∆ABC to ∆DEF is , or The similarity ratio of ∆DEF to ∆ABC is , or 2. . 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 Example 2A Continued Step 1 Identify pairs of congruent angles. All s of a rect. are rt. s and are . A E, B F, C G, and D H. Step 2 Compare corresponding sides. Thus the similarity ratio is , and rect. ABCD ~ rect. EFGH. Example 2B: Identifying Similar Polygons Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. Example 2B Continued Step 1 Identify pairs of congruent angles. P R and S W isos. ∆ Step 2 Compare corresponding angles. mW = mS = 62° mT = 180° – 2(62°) = 56° Since no pairs of angles are congruent, the triangles are not similar. Check It Out! Example 2 Determine if ∆JLM ~ ∆SPN. If so, write the similarity ratio and a similarity statement. Step 1 Identify pairs of congruent angles. N M, L P, S J Check It Out! Example 2 Continued Step 2 Compare corresponding sides. Thus the similarity ratio is , and ∆LMJ ~ ∆PNS. 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. 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. 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. Check It Out! Example 3 Continued 1.25(36.25) = x(9) Cross Products Prop. 45.3 = 9x 5x Simplify. Divide both sides by 9. The length of the model is approximately 5 inches. CW pg 285 prob 6 – 11 all; prob #12- 34 even Due at end of class 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 Lesson Quiz: Part II 3. Tell whether the following statement is sometimes, always, or never true. Two equilateral triangles are similar. Always