1. Problem Solving in Geometry with Proportions

2. Additional Properties of ProportionsIF 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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3. Ex. 1: Using Properties of ProportionsIF 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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4. Ex. 1: Using Properties of ProportionsIF p 3 =
Simplify r 5
 The statement is true.
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5. Ex. 1: Using Properties of Proportionsa 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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6. 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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7. Cross Product Property Divide each side by 20.
Solution
AB = AC
BD CE
16 = 30 – 10 x
16 = 20 x
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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8. a x x b 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 = √a ∙ b which is a positive number. = x b
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9. Geometric Mean Example
For example, the geometric mean of 8 and 18 is 12, because 8 12 = 12 18
and also because x = √8 ∙ 18 = x = √144 = 12
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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.
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11. Write proportion 210 x = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420
Solution:
The geometric mean of 210 and 420 is 210√2, or about 297mm.
210 x
Write proportion = x
420
X2 = 210 ∙ 420
X = √210 ∙ 420
X = √210 ∙ 210 ∙ 2
X = 210√2
Cross product property
Simplify
Factor
Simplify
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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.
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13. Ex. 4: Solving a proportion
MODEL BUILDING. A scale model of the Titanic is inches long and inches wide. The Titanic itself was 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 = in
Length of Titanic = feet
Length of model ship = in.
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14. Reasoning: = = = Write the proportion. Substitute.
Multiply each side by
Use a calculator.
Width of Titanic
Length of Titanic =
Width of model
Length of model
x feet
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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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.
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