Upload
derrick-melton
View
242
Download
4
Tags:
Embed Size (px)
Citation preview
Proportions
What are proportions?
- If two ratios are equal, they form a proportion. Proportions can be used in geometry when working with similar figures.
What do we mean by similar?
- Similar describes things which have the same shape but are not the same size.
1
248
= 1:3 = 3:9
ExamplesThese two stick figures are similar. As you can see both are the same shape. However, the bigger stick figure’s dimensions are exactly twice the smaller.
So the ratio of the smaller figure to the larger figure is 1:2 (said “one to two”). This can also be written as a fraction of ½.
A proportion can be made relating the height and the width of the smaller figure to the larger figure:
2 feet
4 feet
8 feet
4 feet4 ft
2 ft=
8 ft
4 ft
Solving Proportion Problems
First, designate the unknown side as x. Then, set up an equation using proportions. What does the numerator represent? What does the denominator represent?
Then solve for x by cross multiplying:
2 feet
4 feet
8 feet
? feet
4 ft
2 ft=
8 ft
x ft
4x = 16
X = 4
Similar Shapes
In geometry similar shapes are very important. This is because if we know the dimensions of one shape and one of the dimensions of another shape similar to it, we can figure out the unknown dimensions.
Proportions and Triangles
What are the unknown values on these triangles?
16 m
20 m
4 m
3 m
x m
y m
First, write proportions relating the two triangles.
4 m
16 m=
3 m
x m
4 m
16 m=
y m
20 m
Solve for the unknown by cross multiplying.
4x = 48
x = 12
16y = 80
y = 5
Solving for the Building’s Height
Here is a sample calculation for the height of a building:
48 feet
4 feet
3 feet
yardstick
building
x feetx ft
3 ft=
48 ft
4 ft
4x = 144
x = 36
The height of the building is
36 feet.
Ex: The dosage of a certain medication is 2 mg for every 80 lbs of body weight. How many milligrams
of this medication are required for a person who weighs 220 lbs?
Use this rate to determine the dosage for 220-lbs by setting up a proportion (match units)
lbs80mg2
Let x = required dosage
=220 lbs x mg 2(220) = 80x
440 = 80x x = 5.5 mg
Ex: To determine the number of deer in a game preserve, a forest ranger catches 318 deer, tags
them, and release them. Later, 168 deer are caught, and it is found that 56 of them are tagged. Estimate how many deer are in the game preserve.
Set up a proportion comparing the initial tag rate to the later catch tag rate
Initial tag rate = later catch tag rate
sizecatchlatertagged#catchlater
sizepopulationtaggedinitially#
deer168tagged56
deerdtagged318
(318)(168) = 56d
53,424 = 56d 56 56
d = 954 deer in the reserve
Ex: An investment of $1500 earns $120 each year. At the same rate, how much additional
money must be invested to earn $300 each year?
What do we need to find?
Let m = additional money to be invested
What is the annual return rate of the investment?
$120 for $1500 investment
What is the desired return?
$300
Set up a proportion comparing the current return rate and the desired return rate
Initial return rate = desired return rate
investmentnewreturndesired
investmentinitialreturninitial
invested)m1500($returndesired300$
invested1500$return120$
120(1500 + m) = (1500)(300)
180,000 + 120m = 450,000
120m = 270,000 m = $2250 additional needs to be invest new investment = $1500 + $2250 = $3750
Divide by 120
Ex: A nurse is to transfuse 900 cc of blood over a period of 6 hours. What rate would the nurse
infuse 300 cc of blood?
What do we need to find?The rate of infusion for 300 cc of blood
What is the rate of transfusion?
900 cc of blood in 6 hoursSet up a proportion comparing the rate of tranfusion
to the desired rate of infusion
But to set up the proportion we need to know how long it takes to insfuse 300 cc of
blood Let h = hours required
hourshcc300
hours6cc900
proportion comparing the rate of tranfusion to the desired rate of infusion
900h = (6)(300) 900h = 1800 h = 2 hours
Therefore, it will take 2 hours to insfuse 300 cc of blood
New insfusion rate = 300 cc / 2 hours
hours2cc300
hours1cc150
150 cc/hour is the insfusion rate
For Polygons to be Similar
corresponding angles must be congruent,
andcorresponding sides must
be proportional
(in other words the sides must have lengths that form equivalent ratios)
Congruent figures have the same size and shape. Similar figures have the same
shape but not necessarily the same size. The two figures below are similar. They have the same shape but not the same
size.
Let’s look at the two triangles we looked at earlier to see if they
are similar.
Are the corresponding angles in the two
triangles congruent?
Are the corresponding sides proportional?
(Do they form equivalent ratios)
Just as we solved for variables in earlier
proportions, we can solve for variables to find
unknown sides in similar figures.
Set up the corresponding sides as a proportion and
then solve for x.
Ratios x/12and5/10
x 5
12 10
10x = 60
x = 6
In the diagram we can use proportions
to determine the height of the tree.
5/x = 8/288x = 140
x = 17.5 ft