1. Copyright © Cengage Learning. All rights reserved. Rational Expressions and Equations; Ratio and Proportion 6

2. Copyright © Cengage Learning. All rights reserved. Section 6.8 Proportions and Similar Triangles

3. 3 Objectives Determine whether an equation is a proportion. Solve a proportion. Solve an application using a proportion. Solve an application using the properties of similar triangles. 1 1 2 2 3 3 4 4

4. 4 Proportions and Similar Triangles A statement that two ratios are equal is called a proportion. In this section, we will discuss proportions and use them to solve problems.

5. 5 Determine whether an equation is a proportion 1.

6. 6 Determine whether an equation is a proportion Consider Table 6-3, in which we are given the costs of various numbers of gallons of gasoline. If we find the ratios of the costs to the numbers of gallons purchased, we will see that they are equal. Table 6-3

7. 7 Determine whether an equation is a proportion In this example, each ratio represents the cost of 1 gallon of gasoline, which is $3.80 per gallon. When two ratios such as and are equal, they form a proportion.

8. 8 Determine whether an equation is a proportion Proportions A proportion is a statement that two ratios are equal. Some examples of proportions are and The proportion can be read as “1 is to 2 as 3 is to 6.” The proportion can be read as “7 is to 3 as 21 is to 9.”

9. 9 Determine whether an equation is a proportion The proportion can be read as “8x is to 1 as 40x is to 5.” The proportion can be read as “a is to b as c is to d.”

10. 10 Determine whether an equation is a proportion In the proportion, the numbers 1 and 6 are called the extremes, and the numbers 2 and 3 are called the means. In this proportion, the product of the extremes is equal to the product of the means. 1  6 = 6 and 2  3 = 6

11. 11 Determine whether an equation is a proportion This illustrates a fundamental property of proportions. Fundamental Property of Proportions In any proportion, the product of the extremes is equal to the product of the means. In the proportion, a and d are the extremes, and b and c are the means.

12. 12 Determine whether an equation is a proportion We can show that the product of the extremes (ad) is equal to the product of the means (bc) by multiplying both sides of the proportion by bd to clear the fractions, and observing that ad = bc. ad = bc To eliminate the fractions, multiply both sides by bd, the LCD. Simplify.

13. 13 Determine whether an equation is a proportion Since ad = bc, the product of the extremes equals the product of the means. To determine whether an equation is a proportion, we can check to see whether the product of the extremes is equal to the product of the means.

14. 14 Example Determine whether each equation is a proportion. a. b. Solution: In each part, we check to see whether the product of the extremes is equal to the product of the means. a. The product of the extremes is 3  21 = 63. The product of the means is 7  9 = 63. Since the products are equal, the equation is a proportion:

15. 15 Example – Solution b. The product of the extremes is 8  5 = 40. The product of the means is 3  13 = 39. Since the products are not equal, the equation is not a proportion: cont’d

16. 16 Determine whether an equation is a proportion When two pairs of numbers such as 2 and 3 and 8 and 12, form a proportion, we say that they are proportional. To show that 2 and 3, and 8 and 12, are proportional, we check to see whether the equation is a proportion.

17. 17 Determine whether an equation is a proportion To do so, we find the product of the extremes and the product of the means: 2  12 = 24 3  8 = 24 Since the products are equal, the equation is a proportion, and the numbers are proportional. The product of the extremes The product of the means

18. 18 Solve a proportion 2.

19. 19 Solve a proportion Suppose that we know three of the terms in the proportion To find the unknown value, we multiply the extremes and multiply the means, set them equal, and solve for x: 20x = 5  24 20x = 120 In a proportion, the product of the extremes is equal to the product of the means. Multiply.

20. 20 Solve a proportion x = 6 The unknown value is 6. Divide both sides by 20.

21. 21 Example Solve: Solution: To solve the proportion, we clear the fractions by multiplying the extremes and multiplying the means. Then, we solve for x. 12  x = 18  3 In a proportion, the product of the extremes equals the product of the means.

22. 22 Example – Solution 12x = 54 The solution is cont’d Divide both sides by 12. Multiply.

23. 23 Solve an application using a proportion 3.

24. 24 Solve an application using a proportion When solving application problems, we often need to set up and solve a proportion.

25. 25 Example If 6 avocados cost $10.38, how much will 16 avocados cost? Solution: Let c represent the cost of 16 avocados. The ratios of the numbers of avocados to their costs are equal. 6 avocados is to $10.38 as 16 avocados is to $c. 6 avocados Cost of 6 avocados 16 avocados Cost of 16 avocados

26. 26 Example – Solution 6  c = 10.38(16) 6c = 166.08 c = 27.68 Sixteen avocados will cost $27.68. cont’d Simplify Divide both sides by 6. In a proportion, the product of the extremes is equal to the product of the means.

27. 27 Solve an application using the properties of similar triangles 4.

28. 28 Solve an application using the properties of similar triangles If two angles of one triangle have the same measure as two angles of a second triangle, the triangles will have the same shape. Triangles with the same shape are called similar triangles. In Figure 6-1,  ABC ~  DEF (read the symbol ~ as “is similar to”). Figure 6-1

29. 29 Solve an application using the properties of similar triangles Property of Similar Triangles If two triangles are similar, the lengths of all pairs of corresponding sides are in proportion. In the similar triangles shown in Figure 6-1, the following proportions are true. and

30. 30 Example – Height of A Tree A tree casts a shadow 18 feet long at the same time as a woman 5 feet tall casts a shadow that is 1.5 feet long. Find the height of the tree. Solution: We let h represent the height of the tree. Figure 6-2 shows the triangles determined by the tree and its shadow and the woman and her shadow. Figure 6-2

31. 31 Example – Solution Since the triangles have the same shape, they are similar, and the lengths of their corresponding sides are in proportion. We can find h by solving the following proportion. 1.5h = 5(18) 1.5h = 90 cont’d In a proportion, the product of the extremes is equal to the product of the means. Multiply.

32. 32 Example – Solution h = 60 The tree is 60 feet tall. cont’d