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Understanding Ratios, Proportions, and Geometric Means in Geometry

This section explores key concepts related to ratios and proportions, including their definitions, equivalent ratios, and methods to simplify them. Learn how to apply the concept of proportions and the cross product property to solve real-world problems. Additionally, understand the geometric mean and how it relates to two positive numbers. Through practical examples, such as estimating tree populations in a forest and finding angles in triangles, this section provides a comprehensive understanding of these essential mathematical concepts.

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Understanding Ratios, Proportions, and Geometric Means in Geometry

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  1. GeometrySection 6.1 Ratios, Proportions, and the Geometric Mean

  2. Ratio • The ratio of a to b is • Also written as a:b • Usually expressed in simplest form • 2 ratios that have the same simplified forms are called equivalent ratios

  3. Examples • Simplify the ratio • 24 yards to 3 yards • 150 cm : 6 m • 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.

  4. Using extended ratios • A triangle’s angle measures are in the extended ratio of 1 : 3 : 5. find the measures of the angles.

  5. Proportions An equation that states that the two ratios are equal. Cross product property. In a proportion the product of the extremes equals the product of the means.

  6. Examples • Solve the Proportion

  7. Example • As part of a science project, you need to estimate the number of blue spruce trees in a 50 acre forest. You count 36 trees in 3 acres and notice that the trees seem to be evenly distributed. Estimate the total number of blue spruce trees in the forest.

  8. Geometric Mean • The geometric mean of two positive numbers a and b is the positive number x that satisfies so and

  9. Examples • Find the geometric mean of the two numbers • 12 and 27 • 18 and 54 • 16 and 18

  10. Assignment • Section 6.1 • Page 360 • Problems #4 – 44 even, 50

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