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Problem Solving in Geometry with Proportions

Problem Solving in Geometry with Proportions. Ratio. A ratio is a comparison of two quantities by division The ratio of a and b can be represented three ways: a/b a:b a to b An extended ration compares three values (i.e. a:b:c). Ratio Example. Extended Ratio Example. Proportion.

aaron-velazquez
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Problem Solving in Geometry with Proportions

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  1. Problem Solving in Geometry with Proportions

  2. Ratio • A ratio is a comparison of two quantities by division • The ratio of a and b can be represented three ways: • a/b • a:b • a to b • An extended ration compares three values (i.e. a:b:c)

  3. Ratio Example

  4. Extended Ratio Example

  5. Proportion • A Proportion is an equation that states two proportions are equal • First and last numbers are the Extremes • Middle two numbers are the Means

  6. Cross Products Property

  7. Additional Properties of Proportions IF a b a c , then = = c d b d IF a c a + b c + d , then = = b d b d

  8. Ex. 1: Using Properties of Proportions IF p 3 p r , then = = r 5 6 10 p r Given = 6 10 p 6 a c a b = = = , then b d c d r 10

  9. Ex. 1: Using Properties of Proportions IF p 3 = Simplify r 5  The statement is true.

  10. Ex. 1: Using Properties of Proportions a c Given = 3 4 a + 3 c + 4 a c a + b c + d = = = , then 3 4 b d b d Because these conclusions are not equivalent, the statement is false. a + 3 c + 4 ≠ 3 4

  11. In the diagram Ex. 2: Using Properties of Proportions AB AC = BD CE Find the length of BD. Do you get the fact that AB ≈ AC?

  12. Solution AB = AC BD CE 16 = 30 – 10 x 10 16 = 20 x 10 20x = 160 x = 8 Given Substitute Simplify Cross Product Property Divide each side by 20. So, the length of BD is 8.

  13. Geometric Mean • The geometric mean of two positive numbers a and b is the positive number x such that a x If you solve this proportion for x, you find that x = √a ∙ b which is a positive number. = x b

  14. Geometric Mean Example • For example, the geometric mean of 8 and 18 is 12, because 8 12 = 12 18 and also because x = √8 ∙ 18 = x = √144 = 12

  15. PAPER SIZES. International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x. Ex. 3: Using a geometric mean

  16. Solution: The geometric mean of 210 and 420 is 210√2, or about 297mm. 210 x Write proportion = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420 X = √210 ∙ 210 ∙ 2 X = 210√2 Cross product property Simplify Factor Simplify

  17. Using proportions in real life • In general when solving word problems that involve proportions, there is more than one correct way to set up the proportion.

  18. Ex. 4: Solving a proportion • MODEL BUILDING. A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it? Width of Titanic Length of Titanic = Width of model Length of model LABELS: Width of Titanic = x Width of model ship = 11.25 in Length of Titanic = 882.75 feet Length of model ship = 107.5 in.

  19. Write the proportion. Substitute. Multiply each side by 11.25. Use a calculator. Reasoning: Width of Titanic Length of Titanic = Width of model Length of model x feet 882.75 feet = 11.25 in. 107.5 in. 11.25(882.75) x = 107.5 in. x ≈ 92.4 feet So, the Titanic was about 92.4 feet wide.

  20. Note: • Notice that the proportion in Example 4 contains measurements that are not in the same units. When writing a proportion in unlike units, the numerators should have the same units and the denominators should have the same units. • The inches (units) cross out when you cross multiply.

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