7.1 Ratio and Proportion Objective: Find and simplify the ratio of two numbers

7.1 Ratio and Proportion

Objective: Find and simplify the ratio of two numbers.

Computing Ratios

• If a and b are two quantities that are measured in the same units, then the ratio of a to b is a/b. The ratio of a to b can also be written as a:b. Ratios must be SIMPLIFIED. For instance, the ratio of 6:8 is usually simplified to 3:4. (You divided by 2)

Ex. 1: Simplifying Ratios

• Simplify the ratios:

a. 12 cm b. 6 ft c. 9 in.

4 cm 18 ft 18 in.

Ex. 1: Simplifying Ratios

• Simplify the ratios:a. 12 cm b. 6 ft

4 m 18 in

Solution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible.

Ex. 1: Simplifying Ratios

• Simplify the ratios:

a. 12 cm

4 m

12 cm 12 cm 123

4 m 4∙100cm 400 100

Ex. 1: Simplifying Ratios

• Simplify the ratios:

b. 6 ft

18 in

6 ft 6∙12 in 72 in. 4 4

18 in 18 in. 18 in. 1

Ex. 3: Using Extended Ratios

• The measures of the angles in ∆JKL are in the extended ratio 1:2:3. Find the measures of the angles.

• Begin by sketching a triangle. Then use the extended ratio of 1:2:3 to label the measures of the angles as x°, 2x°, and 3x°.

J

K

L

x°

2x°

3x°

Solution:

Statement

x°+ 2x°+ 3x° = 180°

6x = 180

x = 30

Reason

Triangle Sum Theorem

Combine like terms

Divide each side by 6

So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.

Using Proportions

• An equation that equates two ratios is called a proportion. For instance, if the ratio of a/b is equal to the ratio c/d; then the following proportion can be written:

= Means Extremes

The numbers a and d are the extremes of the proportions. The numbers b and c are the means of the proportion.

Properties of proportions

1. CROSS PRODUCT PROPERTY. The product of the extremes equals the product of the means.

If

= , then ad = bc

Properties of proportions

2. RECIPROCAL PROPERTY. If two ratios are equal, then their reciprocals are also equal.

If = , then = ba

To solve the proportion, you find the value of the variable.

Ex. 5: Solving Proportions

4x

57=

Write the original proportion.

Reciprocal prop.

Multiply each side by 4

Simplify.

x4

75=

4 4

x = 285

Ex. 5: Solving Proportions

3y + 2

2y=

Write the original proportion.

Cross Product prop.

Distributive Property

Subtract 2y from each side.

3y = 2(y+2)

y = 4

3y = 2y+4

7.2 Similar Polygons

Objective: To identify and apply similar polygons.

Identifying similar polygons• When corresponding angles of two

polygons are congruent and the lengths of corresponding sides are proportional the two polygons are called similar polygons.

• The symbol ~ is used to indicate similarity. So, ABCD ~ EFGH.

Similar polygons

B

C

AD

F

G

EH

AB= =

EF

BC

FG=

CDGH

DAHE

AB=

EF

Ex. 1: Writing Similarity Statements

• Pentagons JKLMN and STUVW are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.

J K

L

M

N

S T

U

V

W

Ex. 1: Writing Similarity Statements

J K

L

M

N

S T

U

V

W

Because JKLMN ~ STUVW, you can write J S, K T, L U, M V AND N W.

You can write the proportionality statement as follows:

KL

TU=

JK=

ST

MN

VW=

LM=

UV

NJ

WS

Ex. 2: Comparing Similar Polygons

• Decide whether the figures are similar. If they are similar, write a similarity statement.

15

12

9

6X

W

Z

Y

10

8

6

4Q

P

S

R

15

12

9

6X

W

Z

Y

10

8

6

4Q

P

S

RSOLUTION:

As shown, the corresponding angles of WXYZ and PQRS are congruent. Also, the corresponding side lengths are proportional.

WX

PQ=

15

10=

3

2

XY

QR=

6

4=

3

2

YZ

RS=

9

6=

3

2

WX

PQ=

15

10=

3

2So, the two figures are similar and you can write WXYZ ~ PQRS.

Using similar polygons in real life

• If two polygons are similar, then the ratio of lengths of two corresponding sides is called the scale factor. In Example 2 on the previous page, the common ratio of is the scale factor of WXYZ to PQRS.

3

2

Ex. 4: Using similar polygons

• The rectangular patio around a pool is similar to the pool as shown. Calculate the scale factor of the patio to the pool, and find the ratio of their perimeters.

16 ft 24 ft32 ft

48 ft

• Because the rectangles are similar, the scale factor of the patio to the pool is 48 ft: 32 ft. , which is 3:2 in simplified form.

• The perimeter of the patio is 2(24) + 2(48) = 144 feet and the perimeter of the pool is 2(16) + 2(32) = 96 feet The ratio of the perimeters is

16 ft 24 ft32 ft

48 ft144

96

3

2, or

• Theorem 8.1: If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding parts.

• If KLMN ~ PQRS, then

P

S

Q

RK

N

L

M

KL + LM + MN + NK

PQ + QR + RS + SP=

KLPQ

LMQR

MNRS

NKSP

= = =

Ex. 1: Writing Proportionality Statements

• In the diagram, ∆BTW ~ ∆ETC.

a. Write the statement of proportionality.

12

203

T

B W

E C

79°

34°

ET

BT

TC

TW

CE

WB= =

Ex. 1: Writing Proportionality Statements

• In the diagram, ∆BTW ~ ∆ETC.

b. Find mTEC.B TEC, SO

mTEC = 79°

12

203

T

B W

E C

79°

34°

Ex. 1: Writing Proportionality Statements

• In the diagram, ∆BTW ~ ∆ETC.

c. Find ET and BE.

12

203

T

B W

E C

79°

34°

CEWB

ETBT

=

312

ET20

=

3(20)12

= ET

ET=5

Write proportion.

Substitute values.

Multiply each side by 20.

Simplify.

Because BE = BT – ET, BE = 20 – 5 = 15. So, ET is 5 units and BE is 15 units.