Problem Solving in Geometry with

Proportions

Slide #2

Additional Properties of Proportions

ab =

cd

ac =

bd, then

ab =

cd

a + b

b = d, then

IF

IFc + d

Slide #3

Ex. 1: Using Properties of Proportions

p6 =

r10

pr =

35, then

IF

p6 =

r10

pr =

610

Given

a

b=

c

d , thena

c=

b

d

Slide #4

Ex. 1: Using Properties of Proportions

pr =

35

IF

Simplify

The statement is true.

Slide #5

Ex. 1: Using Properties of Proportions

a3 =

c4

Given

a + 3

3 = 4a

b=

c

d , thena + b

b=

c + d

d

c + 4

a + 3

3 ≠ 4c + 4 Because these

conclusions are not equivalent, the statement is false.

Slide #6

Ex. 2: Using Properties of Proportions

In the diagram

AB=

BD

AC

CE

Find the length of

BD.

10

30

x

16

A

DE

B C

Do you get the fact that AB ≈ AC?

Slide #7

Solution

AB = AC

BD CE

16 = 30 – 10

x 10

16 = 20

x 10

20x = 160

x = 8

Given

Substitute

Simplify

Cross Product Property

Divide each side by 20.

10

30

x

16

A

DE

B C

So, the length of BD is 8.

Slide #8

Geometric Mean

The geometric mean of two positive numbers a and b is the positive number x such that

ax =

xb

If you solve this proportion for x, you find that x = √a ∙ b which is a positive number.

Slide #9

Geometric Mean Example

For example, the geometric mean of 8 and 18 is 12, because

and also because x = √8 ∙ 18 = x = √144 = 12

812 = 18

12

Slide #10

Ex. 3: Using a geometric mean

PAPER SIZES. International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x.

420 mmA3

x

x

210 mm

A4

Slide #11

Solution:

420 mmA3

x

x

210 mm

A4

210

x=

x

420

X2 = 210 ∙ 420

X = √210 ∙ 420

X = √210 ∙ 210 ∙ 2

X = 210√2

Write proportion

Cross product property

Simplify

Simplify

Factor

The geometric mean of 210 and 420 is 210√2, or about 297mm.

Slide #12

Using proportions in real life

In general when solving word problems that involve proportions, there is more than one correct way to set up the proportion.

Slide #13

Ex. 4: Solving a proportion

MODEL BUILDING. A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it?

Width of Titanic Length of Titanic

Width of model Length of model=

LABELS:

Width of Titanic = x

Width of model ship = 11.25 in

Length of Titanic = 882.75 feet

Length of model ship = 107.5 in.

Slide #14

Reasoning:

Write the proportion.

Substitute.

Multiply each side by 11.25.

Use a calculator.

Width of Titanic Length of Titanic

Width of model Length of model

x feet 882.75 feet

11.25 in. 107.5 in.

11.25(882.75)

107.5 in.

=

=x

x ≈ 92.4 feet

=

So, the Titanic was about 92.4 feet wide.

Slide #15

Note:

Notice that the proportion in Example 4 contains measurements that are not in the same units. When writing a proportion in unlike units, the numerators should have the same units and the denominators should have the same units.

The inches (units) cross out when you cross multiply.