Geometry

Sources:

Discovering Geometry (2008) by Michael Serra

Geometry (2007) by Ron Larson

What is the the relationship between two geometric figures that look “alike” but have different “sizes”?

Working with similar figures involves ratio and proportion.

Review Ratio and Proportion.

Understand Similar Polygons

Exercises

Homework: WS on Similar Triangles

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)

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)

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.

ab

The order in which a ratio is written or stated is important.

The ratio of Macbooks to iPod Touch’s sold last Christmas is:

2:9

Macbooks 2Touch 9

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:a cb d

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.

a (first term) c (third term)b (second term) d (fourth term)

a:b c:d

extremes

means

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:

a cb d

a x k cb x k d

a cb d

ad bca cb d

Also used in...

... Scale modeling

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.

These are Similar Polygons.

14 16

10

2421

152

4

2

1

3

6

sim⋅i⋅lar[sim-uh-ler]  

–adjective1.having a likeness or resemblance,

esp. in a general way: 

example: two similar houses.

(dictionary.com)

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?

TRIVIA: The symbol for

similarity “~” is called a “tilde”, or sometimes

“twiddle”.

Rectangle QUIL is SIMILAR to Rectangle SETH.

Rectangle QUIL is SIMILAR to Rectangle SETH.

9

21

3

7Q U

IL

S E

TH

The SCALE FACTOR is 3.

14 16

10 X

Y

Z

2421

15A

B

C

The SCALE FACTOR is 3/2.

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

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

using proportionality.