Chapter 6.1 Notes: Ratios, Proportions, and the Geometric Mean Goal: You will solve problems by writing and solving proportions.

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Presentation transcript:

Chapter 6.1 Notes: Ratios, Proportions, and the Geometric Mean Goal: You will solve problems by writing and solving proportions.

Ratios If a and b are two numbers or quantities and b ≠ 0, then the ratio of a to b is The ratio of a to b can also be written as a:b.

A D 2m 2m 1m 1m B 2 m C E 1 m F The ratio of side length in ∆ABC to side length in ∆DEF can be written as or 2:1. Ratios are expressed in simplest form.

Two ratios that have the same simplified form are called equivalent ratios. Ex.1: Simplify the ratio. a. 64 m : 6 m b. c. 150 cm : 6 m

Ex.2: You are planning to paint a mural on a rectangular wall. You know that the perimeter of the wall is 484 feet and that the ratio of its length to its width is 9:2. Find the area of the wall.

Ex.3: The measures of the angles in ∆CDE are in the extended ratio of 1:3:5. Find the measures of the angles. Ex.4: The perimeter of a room is 48 feet and the ratio of its length to its width is 7:5. Find the length and width of the room. Ex.5: The measures of the angles of ∆RST are in the extended ratio of 2:3:4. Find the measures of the angles.

Proportions An equation that states that two ratios are equal is called a proportion. The numbers b and c are the means of the proportion. The numbers a and d are the extremes of the proportion.

Cross Products Property: In a proportion, the product of the extremes equals the product of the means. Ex.6: Solve the proportion. a. b.

Ex.7: Solve the proportion. a. b. Geometric Mean The geometric mean of two positive numbers a and b is the positive number x that satisfies

Ex.8: Find the geometric mean of 24 and 48. Find the geometric mean of the two numbers. Ex.9: 12 and 27 Ex.10: 18 and 54 Ex.11: 16 and 18

Ex.12: As part of an environmental study, you need to estimate the number of trees in a 150 acre area. You count 270 trees in a 2 acre area and you notice that the trees seem to be evenly distributed. Estimate the total number of trees. Ex.13: In example 12, suppose you count 390 trees in a 3 acre area of the 150 acre area. Make a new estimate of the total number of trees.