Ratios and Proportions
Scale Factor
A scale factor is a number which scales, or multiplies, some quantity.
Scale factors are used when certain real world numbers need to be represented
on another scale in order to fit a required format.
Example
Scale
1 in. = 1,000 mi
Example
A photocopy machine reduces pictures in perfect proportion to the actual photo.
A copy of a rectangular picture that has a length of 8” and width of 12” is made
and the new width of 9”. What is the new length?
Answer:
To solve scale factor problems, you are going to use proportions. You can set up
the proportion several different ways but the key is to stay consistent.
old length new length
or width = new width
old
old width
old length
=
new length new width
Choosing the proportion on the right, we would plug in the values that we know.
old length = 8” old width = 12” new width = 9”
There is no value for “new length” (since that is what we are trying to find).
Therefore, we will substitute a variable to represent “new length” (n).
8 n
=
12by 9cross multiplying.
Then solve the proportion
72 = 12n Æ n = 6
Therefore, the new length = 6”.
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Similar triangles have the following two characteristics:
Corresponding sides are proportional
Corresponding angles have the same measures.
E
Example
B
3
A
3
4
6
2
C
F
6
D
∆ABC ~ ∆DEF
1.
2.
3.
4.
Steps for finding Missing Lengths
Set up proportion.
Cross multiply.
E
Solve for missing side.
Check to make sure answer makes sense.
Example
8
Given ∆ABC ~ ∆DEF, find the value for x.
Answer:
1. Set up proportion.
small big
4 8
Æ
=
=
small big
x 6
F
8
D
B
2. Cross multiply.
24 = 8x
3. Solve for missing side
x=3
6
4
A
x
4
C
Real-Life Problem
A building that is 50 feet tall casts a 20 foot shadow on the ground at a certain
time during the day. What is the length of the shadow on the ground cast by a
10-foot tree?
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Answer:
1. Create similar triangles.
Since both objects cast shadows, we will let the shadow be one pair of
corresponding sides. The objects that are casting shadows will be another pair of
corresponding sides. The angle between the building and the ground and the
tree and the ground are congruent because we are assuming the building and
tree are perpendicular to the ground.
Building
50 ft
Tree
10 ft
Tree
shadow
x ft
Building
shadow
20 ft
2. Set up a proportion.
building height
tree height
=
Æ
building shadow tree shadow
50 10
=
20 x
3. Cross multiply.
50x = 200
4. Solve.
x=4
So, the tree’s shadow has a length of 4 ft.
Here are some examples of when proportions are used in Geometry:
The Geometric Mean Altitude Theorem states that the length of the altitude
in a right triangle is the geometric mean between the lengths of the two
segments of the hypotenuse.
S
x
a
Q
T
a y
=
x
a
y
U
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The Geometric Mean Leg Theorem states that the length of a leg in a right
triangle is the geometric mean between the length of the hypotenuse and the
segment of the hypotenuse adjacent to that leg.
S
x
T
Q
b
y
=
x+y b
y
U
b
The Altitudes of Similar Triangles Theorem states that if two triangles are
similar, the lengths of the corresponding altitudes are proportional to the lengths
of corresponding sides.
a
c
b
a b
=
d c
d
The Medians of Similar Triangles Theorem states that if two triangles are
similar, the lengths of the corresponding medians are proportional to the lengths
of corresponding sides.
d
a
b
c
a b
=
d c
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The Angle Bisectors of Similar Triangles Theorem states that if two triangles
are similar, the lengths of the corresponding angle bisectors are proportional to
the lengths of corresponding sides.
a
b
c
d
a b
=
d c
The Side-Splitting Theorem states that if a line is parallel to one side of a
triangle and intersects the other two sides, then it divides the sides into
segments of proportional lengths.
a
b
c
d
a c
=
b d
Example:
Solve for x.
Use the Side Splitting Theorem:
7
6
Set up the proportion:
=
14 x
Cross multiply:
7x = 84
Divide both sides by 7:
x = 12
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7
14
6
x