Ratio and Proportion

1. Ratio and Proportion Geometry Sources: Discovering Geometry (2008) by Michael Serra Geometry (2007) by Ron Larson

2. Working with similar figures involves ratio and proportion. What is the the relationship between two geometric figures that look “alike” but have different “sizes”?

3. Objectives Review Ratio and Proportion. Understand Similar Polygons Exercises Homework: WS on Similar Triangles

4. Understanding Ratio A ratio is a way that quantities can be divided or shared. It is an expression that compares two numbers by division. You are already familiar with ratios that are used in everyday situations. You use a ratio in art when you mix paint colors, on a map when you read the map scales, and in cooking when you use the ratios of ingredients. (http://pilotmath.com/teachersampler/courses/essentials/concept_capsule/section0_slide2.html)

5. Understanding Ratio Use 1 measure nectar syrup to 5 measures club soda Use 1 shovel of cement to 3 shovels of sand Use 3 parts blue paint to 1 part white (http://pilotmath.com/teachersampler/courses/essentials/concept_capsule/section0_slide2.html)

6. Understanding Ratio We define ratio as : An expression that compares two quantities by division. If a and b are 2 numbers, where , then the ratio of a to b is written as . It can also be written as a:b or a is to b.

7. Remember! 2:9 • The order in which a ratio is written or stated is important. • The ratio of Macbooks to iPod Touch’s sold last Christmas is:

8. Understanding Proportion A proportion is a mathematical sentence that states that two ratios are equivalent. It is a statement of equality between 2 ratios. It is used in solving problems that involve comparison of similar objects or situations. Example:

9. Understanding Proportion means extremes Each number in a proportion is called a term. The second and third terms are called the means. The first and fourth terms are called the extremes of the proportion.

10. Solving for Unknowns in Proportions A proportion can be viewed as a multiplicative relationship. In proportional situations the quantities between or across measure spaces always are related by multiplication. Example 1: Example 2:

11. Proportions Also used in... ... Scale modeling

12. Let’s Try These! 1. In a triangle, each side measures 12 cm, 16 cm, and 18 cm, respectively. In lowest terms, find the ratios of the lengths of the sides. 2. The ratio of two supplementary angles is 2 to 3. Find the measure of each angle.

13. So what does “SIMILAR” mean? These are Similar Polygons. 1 16 14 15 2 10 3 2 21 24 4 6

14. So what does SIMILAR mean? sim⋅i⋅lar [sim-uh-ler]  –adjective 1.having a likeness or resemblance, esp. in a general way:  example: two similar houses. (dictionary.com)

15. So what does SIMILAR mean? According to Discovering Geometry by M. Serra, figures that have the same shape but not necessarily the same size are similar. Can we refine this definition?

16. Correspondence in Similar Polygons Rectangle QUILis SIMILAR to Rectangle SETH. TRIVIA: The symbol for similarity “~” is called a “tilde”, or sometimes “twiddle”.

17. Correspondence in Similar Polygons Rectangle QUILis SIMILAR to Rectangle SETH. 7 U Q 3 21 E I S L The SCALE FACTOR is 3. 9 T H

18. Correspondence in Similar Polygons Y 16 14 15 C A X Z 10 21 24 The SCALE FACTOR is 3/2. B

19. Similarity We define similar polygons in Geometry as polygons having the same shape; specifically having congruent corresponding angles and proportional corresponding sides.

20. Exercise: • Polygon MAUI is SIMILAR to polygon KYLE. • Express in symbol form. • Identify corresponding congruent angles. • Express corresponding lengths of sides using proportionality.