1. Problem Solving in Geometry with Proportions

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

3. Ratio Example

4. Extended Ratio Example

5. 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

6. Cross Products Property

7. 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

8. 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

9. Ex. 1: Using Properties of ProportionsIF p 3 =
Simplify r 5
 The statement is true.

10. 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

11. 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?

12. 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.

13. 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

14. 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

15. 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.

16. 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

17. 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.

18. 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.

19. 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.

20. 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.