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8.2 Problem Solving in Geometry with Proportions Geometry Mrs. Spitz Spring 2005

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Slide #2 Objectives/Assignment Use properties of proportions Use proportions to solve real-life problems such as using the scale of a map. Pp all Reminder: Quiz after 8.3. Ch. 8 Definitions due Ch. 8 Postulates/Theorems due

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Slide #3 Using the properties of proportions In Lesson 8.1, you studied the reciprocal property and the cross product property. Two more properties of proportions, which are especially useful in geometry, are given on the next slides. You can use the cross product property and the reciprocal property to help prove these properties.

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Slide #4 Additional Properties of Proportions a b = c d a c = b d, then a b = c d a + b b = d, then IF c + d

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

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Slide #6 Ex. 1: Using Properties of Proportions p r = 3 5 IF Simplify The statement is true.

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Slide #7 Ex. 1: Using Properties of Proportions a 3 = c 4 Given a = 4 a b = c d, then a + b b = c + d d c + 4 a c + 4 Because these conclusions are not equivalent, the statement is false.

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Slide #8 Ex. 2: Using Properties of Proportions In the diagram AB = BD AC CE Find the length of BD. Do you get the fact that AB AC?

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Slide #9 Solution AB = AC BD CE 16 = 30 – 10 x = 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.

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Slide #10 Geometric Mean The geometric mean of two positive numbers a and b is the positive number x such that a x = x b If you solve this proportion for x, you find that x = a b which is a positive number.

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Slide #11 Geometric Mean Example For example, the geometric mean of 8 and 18 is 12, because and also because x = 8 18 = x = 144 = = 18 12

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Slide #12 Ex. 3: Using a geometric mean 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.

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Slide #13 Solution: 210 x = x 420 X 2 = X = X = X = 2102 Write proportion Cross product property Simplify Factor The geometric mean of 210 and 420 is 2102, or about 297mm.

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Slide #14 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.

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Slide #15 Ex. 4: Solving a proportion MODEL BUILDING. A scale model of the Titanic is inches long and inches wide. The Titanic itself was feet long. How wide was it? Width of TitanicLength of Titanic Width of modelLength of model = LABELS: Width of Titanic = x Width of model ship = in Length of Titanic = feet Length of model ship = in.

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Slide #16 Reasoning: Write the proportion. Substitute. Multiply each side by Use a calculator. Width of TitanicLength of Titanic Width of modelLength of model x feet feet in in (882.75) in. = = x x 92.4 feet = So, the Titanic was about 92.4 feet wide.

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Slide #17 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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