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?

15

10

ft

yds

10

300

m

cm

7

4

days

weeks

1 3 1

5 10

x x

6 15

3 4x x

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

Outcomes• I will be able to:

• 1) Simplify ratios

• 2) Solve proportions

• 3) Use properties of proportions

• 4) Define and use the geometric mean

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)

in

ft

12

3

m

cm

50

1000

Parts of a Proportion

d

c

b

a

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 extremesAnd the numbers on the inside are the means

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

Proportions

extremes

means So, ad = bc

reciprocals

So, c

d

a

b

More Properties of Proportions

• Additional Properties of Proportions

• 3) If , then

• 4) If , then

a c

b d a b

c d

a c

b d a b c d

b d

Examples

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

Examples

1st: Label everything we know

2nd: Use that to look for other things

3rd: Use the proportion they gave us

25

60100

75

25

7560

xx = 20

the ratio of two figures, is theratio of corresponding parts

Geometric Mean

a x

x b

ba

***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, thatnumber goes in place of x in the denominator and numerator.

Examples

• 1) Find the geometric mean between 4 and 25

• 2) Twelve is the geometric mean between 8 and what other number

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.

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?

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?

Independent Practice

• Take a look at the 8.2 IP…

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

~

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:

XY YZ XZ

VU UW VW

X

Y Z

V

U W

Examples

, , , ,A W B V C X D Y E Z AB BC CD DE AE

VW WX XY YZ ZV

Examples

Examples

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

Theorem 8.1

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

KL LM MN NK KL

PQ QR RS SP PQ

Example