Download Proportions

Proportions
Proportions
What are proportions?
- If two ratios are equal, they form a proportion.
Proportions can be used in geometry when working with
similar figures.
1
4
2
=
8
1:3 = 3:9
What do we mean by similar?
- Similar describes things which have the same shape
but are not the same size.
Examples
These 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.
8 feet
4 feet
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.
2 feet
A proportion can be made
relating the height and the
width of the smaller figure to
the larger figure:
4 ft
2 ft
=
8 ft
4 ft
4 feet
Solving Proportional Problems
So how do we use
proportions and similar
figures?
8 feet
4 feet
Using the previous
example we can show
how to solve for an
unknown dimension.
2 feet
? feet
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?
4 ft
8 feet
8 ft
=
2 ft
x ft
Then solve for x by cross
multiplying:
4x = 16
X=4
4 feet
2 feet
? feet
Try One Yourself
8 feet
Knowing these two stick
figures are similar to
each other, what is the
ratio between the
smaller figure to the
larger figure?
12 feet
4 feet
x feet
Set up a proportion.
What is the width of the
larger stick figure?
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.
Triangle and Angle Review
Today we will be working with
right triangles. Recall that one of
the angles in a right triangle
equals 90o. This angle is
represented by a square in the
corner.
90o angle
To designate equal angles we will
use the same symbol for both
angles.
equal angles
Proportions and Triangles
What are the unknown values on these triangles?
First, write proportions relating the
two triangles.
20 m
4m
16 m
16 m
3m
=
xm
ym
4m
=
16 m
20 m
Solve for the unknown by cross
multiplying.
xm
ym
4m
3m
4x = 48
16y = 80
x = 12
y=5
Triangles in the Real World
Do you know how tall your school building is?
There is an easy way to find out using right triangles.
To do this create two similar triangles
using the building, its shadow, a
smaller object with a known height
(like a yardstick), and its shadow.
The two shadows can be measured,
and you know the height of the yard
stick. So you can set up similar
triangles and solve for the height of
the building.
Solving for the Building’s Height
Here is a sample calculation for
the height of a building:
x ft
3 ft
=
48 ft
building
x feet
4 ft
48 feet
4x = 144
x = 36
The height of the building is 36
feet.
yardstick
3 feet
4 feet
Accuracy and Error
Do you think using proportions to calculate the
height of the building is better or worse than
actually measuring the height of the building?
Determine your height by the same technique
used to determine the height of the building. Now
measure your actual height and compare your
answers.
Were they the same? Why would there be a
difference?