Problem Solving in Geometry with Proportions

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Problem Solving in Geometry with Proportions
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Ratio

• A ratio is a comparison of two quantities by
  division
• The ratio of a and b can be represented three
  ways
• a/b
• ab
• a to b
• An extended ration compares three values (i.e.
  abc)

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Ratio Example
4
Extended Ratio Example
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Proportion

• A Proportion is an equation that states two
  proportions are equal
• First and last numbers are the Extremes
• Middle two numbers are the Means

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Cross Products Property
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Additional Properties of Proportions
IF
a
b
a
c
,
then


c
d
b
d
IF
a
c
a b
c d
,
then


b
d
b
d
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Ex. 1 Using Properties of Proportions
IF
p
3
p
r
,
then


r
5
6
10
p
r
Given

6
10
p
6
a
c
a
b



, then
b
d
c
d
r
10
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Ex. 1 Using Properties of Proportions
IF
p
3

Simplify
r
5
? The statement is true.
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Ex. 1 Using Properties of Proportions
a
c
Given

3
4
a 3
c 4
a
c
a b
c d



, then
3
4
b
d
b
d
Because these conclusions are not equivalent, the
statement is false.
a 3
c 4
?
3
4
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Ex. 2 Using Properties of Proportions

• In the diagram

AB
AC

BD
CE
Find the length of BD.
Do you get the fact that AB AC?
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• 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.

?So, the length of BD is 8.
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?Geometric Mean?

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

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

x
b
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Geometric Mean Example

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

8
12

12
18
and also because x v8 18 x v144 12
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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.

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• Solution

The geometric mean of 210 and 420 is 210v2, or
about 297mm.
210
x
Write proportion

x
420
X2 210 420 X v210 420 X v210 210
2 X 210v2
Cross product property
Simplify
Factor
Simplify
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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.

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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.
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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)

x
107.5 in.
x 92.4 feet
?So, the Titanic was about 92.4 feet wide.
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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.