The document discusses ratios and proportions, providing examples of how ratios are used in art, architecture, and scale models. It defines ratios and extended ratios, and provides examples of writing ratios and using extended ratios to solve problems involving the sides of triangles given the ratios of the sides and the perimeter. Students should complete homework problems 1, 4, 5, 9, and 10 on pages 284-285 related to ratios and proportions.
5. Why is it important? 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.
7. Why do artists use ratios? Stained-glass artist Louis Comfort Tiffany used geometric shapes in his designs. In a portion of Clematis Skylight , rectangular shapes are used as the background for the flowers and vines. Tiffany also used ratio and proportion in the design of this piece.
8. Why do artists use ratios? What geometric shape did artist Louis Comfort Tiffany use as the background for the flowers and vines? Rectangles What is another example of the use of rectangles in a design? Architects use rectangles in the designs of buildings. What is an example of a scale model? A model airplane.
9.
10.
11.
12. Write Ratios A ratio is a comparison of two quantities. The ratio of a to b can be expressed , where b is not zero. This ratio can also be written a : b . a b
13. Study Tip Ratios should be written in simplest form. A ratio in which the denominator is 1 is called a unit ratio .
14. Write a Ratio The U.S. Census Bureau surveyed 8,218 schools nationally about their girls’ soccer programs. 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. 32.9 can be written as . So, the ratio for this survey was 32.9 girl soccer players for each school in the survey. 32.9 1 Number of girl soccer players Number of schools = 270,273 8,218 or about 32.9.
15. Write a Ratio ( you try! ) The total number of students who participate in sports programs at Central High School is 520. The total number of students in school is 1850. Find the athlete-to-student ratio to the nearest tenth. So, the ratio for this survey was 0.3 athletes to students. Total # of athletes Total Students = 520 1850 or about 0.3.
16. Extended Ratios 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 .
17. Extended Ratios in Triangles 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. 10 in. 60 in. 90 in. 100 in. Remember that equivalent fractions can be found by multiplying the numerator and the denominator by the same number. 2:3 = * or . 2 3 x x 2 x 3 x Thus, we can rewrite 4:6:9 as 4 x :6 x :9 x and use those measures for the sides of the triangle. Write an equation to represent the perimeter of the triangle as the sum of the measures of its sides. Use this value of x to find the measures of the sides of the triangle. 4 x = 4(10) or 40 inches. 6 x = 6(10) or 60 inches 9 x = 9(10) or 90 inches. The longest side is 90 inches. The answer is… A B C D 4 x + 6 x + 9 x = 190 Perimeter 19 x = 190 Combine like terms x = 10 Divide each side by 19 9 x 4 x 6 x C
18. Use Extended Ratios ( you try! ) In a triangle, the ratio of the measures of three sides is 5:12:13, and the perimeter is 90 centimeters. Find the measure of the shortest side of the triangle. 15 cm. 18 cm. 36 cm. 39 cm. 5 x = 5(3) or 15 centimeters. 12 x = 12(3) or 36 centimeters. 13 x = 13(3) or 39 centimeters. The shortest side is… A B C D 5 x + 12 x + 13 x = 90 Perimeter 30 x = 90 Combine like terms x = 3 Divide each side by 30 13 x 12 x 5 x A
