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Proportions
• A proportion is an equation that states that two ratios (written as fractions) are equal.
=
Proportions
• The cross product of a proportion are equal.
• We can use this property to solve proportions for a missing quantity.
=
1= 2=
Problem 1:
• What is the solution to the proportion = ?
=5= 4
= ==
Use cross multiplication to solve.
Problem 2:
• What is the solution to the proportion = ?
=
4(𝑏−8) =
5(𝑏+3)
=
Use cross multiplication to solve.
Use the distributive property on both sides of the equation.
Problem 2:
• What is the solution to the proportion = ?
=
==
Use what you know about solving equations to finish the problem.
= 𝑏 𝑏+32 +32
=
Similar Figures
• Figures are similar if the have the same shape, but not necessarily the same size.
• The angles are congruent (measure the same), but the side lengths are not.
∠𝐴∠𝐵
∠𝐶 ∠𝐸∠𝐹
∠𝐺
Similar Figures• Similar is represented by the
symbol . The sides are similar.
• c
• Congruent is represented by the symbol . The angles are congruent.
• E• •
∠𝐴∠𝐵
∠𝐶 ∠𝐸∠𝐹
∠𝐺𝑎 𝑏
𝑐 𝑔
𝑓𝑒
Problem 3:
In the diagram ∆ABC∆DEF. What is AB?
What is the question asking?AB represents a side, so the question is asking you find the length of the side AB.Write in a variable for the side you are trying to find.
B CFE
AD
16
18
12
10
𝑌
Problem 3:
In the diagram ∆ABC∆DEF. What is AB?
You will need to set up a proportion using corresponding sides. If you look at the original question, it tells you that ∆ABC∆DEF. The order the letters are written in have significance. This means that A corresponds to D (∆ABC∆DEF), B corresponds to E (∆ABC∆DEF), and C corresponds to F (∆ABC∆DEF).
B CFE
AD
16
18
12
10
𝑌
Problem 3:
In the diagram ∆ABC∆DEF. What is AB?
Side AB corresponds to side DE.Side BC corresponds to side EF.Side AC corresponds to side DF.
Since DF does not have a length written, it is not useful. Therefore, you will not need to use the correspondence between side AC and side DF.
B CFE
AD
16
18
12
10
𝑌
Problem 3:
In the diagram ∆ABC∆DEF. What is AB?
Now you will set up the proportion using values from the smaller triangle (∆ABC) on one side andvalues from the larger triangle (∆DEF) on the other.B C
FE
AD
16
18
12
10
𝑌
= Notice that corresponds to 10 and 12 corresponds to 16.
Problem 3:
In the diagram ∆ABC∆DEF. What is AB?
Now you will solve by cross multiplication.
=
B CFE
AD
16
18
12
10
𝑌
=
Percents
• A percent standardizes a comparison to a common base of 100.
• Percents can also be written as decimals by dividing.
25%=
=0.25
Percents
• Percent problems can be solved using both fractions or decimals. Key words can help solve them.
• Method 1: Fractions (proportions) what is up percent of down
Problem 4:
125% of what number is 17.5?
We begin by underlining and identifying key words.
125% of what number is 17.5?
KEY WORDSwhat = is = uppercent = of = down
125❑ =❑
❑
Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDSwhat = is = uppercent = of = down% means over
100
125100=
❑❑
Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDSwhat = is = uppercent = of = downThe # or word after
goes down
125100=
❑❑
Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDSwhat = is = uppercent = of = down𝒙
125100=
❑𝑥
Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDSwhat = is = uppercent = of = downThe # or word after
goes up
125100=
17.5𝑥
Problem 4:
125% of what number is 17.5?Now you can solve by cross multiplication.
=125 17.5125
=
𝑥=14
Percents
• Method 2: Decimals • what is = percent convert the number into of multiply a decimal by moving the decimal point two places to the left
Problem 5:
What is 20% of 140?
We begin by underlining and identifying key words. What is 20% of 140?
KEY WORDSwhat : is : =percent : decimal of : multiply
Problem 5:
What is 20% of 140?
What is 20% of 140? =
KEY WORDSwhat : is : =percent : decimal of : multiply
Problem 5:
What is 20% of 140?
What is 20% of 140? = 0.20
KEY WORDSwhat : is : =percent : decimal of : multiply
20 is written as a decimal by moving the decimal point that is implied behind the zero (20.) two places to the left: 0.20
Problem 5:
What is 20% of 140?
What is 20% of 140? = 0.20
KEY WORDSwhat : is : =percent : decimal of : multiply
Problem 5:
What is 20% of 140?
What is 20% of 140? = 0.20140
KEY WORDSwhat : is : =percent : decimal of : multiply