8.2 Problem Solving in Geometry with Proportions

1. 8.2 Problem Solving in Geometry with Proportions Geometry Mrs. Spitz Spring 2005

2. Use properties of proportions Use proportions to solve real-life problems such as using the scale of a map. Pp. 2-32 all Reminder: Quiz after 8.3. Ch. 8 Definitions due Ch. 8 Postulates/Theorems due Objectives/Assignment

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.

4. 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

5. 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

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

7. 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

8. 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?

9. 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.

10. 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

11. 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

12. 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

13. 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

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.

15. 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.

16. 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.

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.