1. Bell Work 1/22/13 1) Simplify the following ratios: a)b)c) 2) Solve the following proportions: a)b) 3) A map in a book has a scale of 1 in = 112 miles, and you measured the state of Indiana to be 1.5 inches wide. How many miles wide is the state of Indiana really?

2. Agenda 1) Bell Work 2) Agenda/Outcomes 3) Homework check 4) Proportion Properties 5) Geometric Mean 6) Proportion word problems ) Begin IP

3. Outcomes I will be able to: 1) Simplify ratios 2) Solve proportions 3) Use properties of proportions 4) Define and use the geometric mean

4. Ratio Review Ratio – a comparison of two quantities in the same units To solve: 1) Convert to the same units(Multiply) 2) Simplify(Reduce) Examples: a) b)

5. Parts of a Proportion Think about each side of this proportion as a ratio. How else could we write these ratios? *a:b and c:d Each proportion has two parts, 1) extremes 2) means *The numbers on the outside of the ratio are the extremes And the numbers on the inside are the means

6. On Your Own Take a few minutes to solve the following. Decide whether or not each statement is true or false. We’ll do #1 together. Cross multiply to verify if the “if” statement and the “then” are equal

7. Proportions extremes means So, ad = bc reciprocals So,

8. More Properties of Proportions Additional Properties of Proportions 3) If, then 4) If, then

9. Examples Use the properties of proportions to verify if the “if” and the “then” can be the same.

10. Examples 1 st : Label everything we know 2 nd : Use that to look for other things 3 rd : Use the proportion they gave us 25 60 100 75 x = 20 the ratio of two figures, is the ratio of corresponding parts

11. Geometric Mean ***x is always the geometric mean ***1) If we are looking for the mean, x will remain in the denominator and numerator. 2) If we are given the mean, that number goes in place of x in the denominator and numerator.

12. Examples 1) Find the geometric mean between 4 and 25 2) Twelve is the geometric mean between 8 and what other number

13. Proportions in real-life Proportions are very useful in real life. Companies often create scale models of their products before constructing larger models. Example 1:An engineer makes model cars so that his 3-inch model represents an 8-foot-long car. (a) What ratio model : car does he use? (Remember to use the same units!) (b) Use the ratio from part (a) to find the height of the model if the car is 5 feet tall.

14. Examples 2. The Titanic was 882 feet and 9 inches long. A model of the ship is 2 feet 6 inches long and 6 inches high. What was the approximate height of the Titanic to the nearest inch?

15. Examples 3. An architect is to design a skyscraper that is 200 feet long, 140 feet wide, and 400 feet tall. She would like to build a model so that the similarity ratio of the model to the building is 1:400. What should the length and width of the model be in inches?

16. Independent Practice Take a look at the 8.2 IP…

17. Similar Figures Similar Polygons: Two polygons such that their corresponding angles are ______________________ and the lengths of corresponding sides are _____________________________. The symbol for “is similar to” is _______. congruent proportional ~

18. Statement of Proportionality Statement of Proportionality: An (extended) equation that relates all of the equal ratios in a polygon. For instance, if we said ∆XYZ ~ ∆VUW, we would have the following statement of proportionality: X YZ V UW

19. Examples

22. Scale Factor Scale Factor - The ratio of the lengths of two corresponding sides of two similar polygons.

23. Theorem 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

24. Example