Download 5” 3” 11” 8.5”

Chapter 6
Proportions and Similarity
6.1 Proportions
Ratios
A ratio is a comparison of two quantities.
The these quantities must be integers,
must be in simplified form and must be of
the same unit of measure.
If comparison is not of the same unit of
measure, then it is called a rate.
Ratios can be written three different ways:
a:b
a to b
a/b
Proportions
Proportions are equations that state two (or
more) ratios are equal.
Cross product rule – the product of the means
equals the product of the extremes.
a c

b d
Product of extremes = ad
Product of means = bc
Cross Product Rule:
ad = bc
Example
3 x  5 13

4
2
2(3x  5)  4(13)
6x 10  52
6x  42
x  7
Cross Product Rule
Distribution
Add 10 to both sides
Divide both sides by 6
Rates
Remember, Rates are like ratios in that
they have integers as numbers and are
written in lowest form.
The thing that is different is that the
numbers have different units of measure.
MPH?
MPG?
$/hr?
Example:
Say you have a model car that has a
wheel diameter of 1” and the wheel
diameter of the real car is 18”. If you
measure the length of the model and you
find it is 9” long, how long is the car?
Set up the ratio of
model wheel/real wheel = model length/real length
Plug in the numbers and go….
6.2 Similar Polygons
Similar Polygons
When two polygons have the same shape but are
not congruent, they may be similar.
By definition, a similar polygon is where all the
corresponding angles are congruent, but all the
ratios of corresponding sides are equal.
If quad ABCD ~ EFGH, then….
A  E , B  F , C  G and
AB BC CD AD



EF FG GH EH
D H
Scale Factor
The ratio of the corresponding sides is called the
scale factor.
The scale factor is a ratio – that means that the
numbers must be integers and the ratio must be
in lowest, simplified form.
If the scale factor is 1 to 2 that means that the
2nd figure is two times larger than the first.
So, for every unit of length in the first figure, if
you multiply it by 2, you’ll get the length of the
corresponding side in the 2nd figure.
Or the length of the first figure = (1/2) length of
the 2nd figure.
Example:
If you have a picture and you “blow it up”
on a copy machine by 20%, what is the
scale factor of original to new?
Blowing up by 20% is the same as
multiplying the new figure by 120% or 1.2.
So, the ratio is 1 to 1.2 which simplifies to
5 to 6.
Is the new figure larger? So, that is why it
is 5 to 6.
Another Example
You have a photo that is 3” x 5” – you want
to “blow it up” as large as it can get to fit
on a 8.5” x 11” of paper, BUT you want to
keep the same aspect of length and width.
How big can it be without cutting any of
the photo off?
What is “Aspect”?
That is keeping the ratio of the H vs L the
same….
Example Continued
3”
8.5”
5”
(3/5) ≠ (8.5/11)
Set up two ratios:
11”
(3/5) = (x/11)
Solving for x and y we get…
and
x = 6.6, y = 14 1/6
(3/5) = (8.5/y) So, y is too big – we’ll cut off something
Continued
6.6”
8.5”
11”
So, 3 x 5 picture gets blown up to 6.6 x 11 to get the
bird as big as it can get w/o cutting things off.
6.3 Similar Triangles
Similar Triangles
Remember when we said that the only
way to prove triangles congruent was to
prove all 3 sets of sides and the 3 sets of
angles were congruent?
Then we told you that there were four
short cuts?
SSS, SAS, ASA and AAS?
Well, there are 3 short cuts for Similar
Triangles.
Similar Triangle Short Cuts
SSS~
This says that when you have the three ratios of
corresponding sides equal, then you have similar
triangles.
SAS~
This says that when you have congruent, included
angles and the ratios of the two sets of sides equal,
then you have similar triangles.
AA
This says that when you have two sets of
corresponding angles congruent, then the triangles
are similar.
SSS ~
B
E
16
12
24
18
A
8
C
D
Is 8/12 = 12/18 = 16/24?
F
12
Yes, SF = 2/3
So ΔABC ~ ΔDEF by SSS ~

Because of Def of Similar Polygons, <A  <D

SAS ~
B
E
12
18
33°
A
8
C
D
33°
Is 8/12 = 12/18?
F
12
Yes, SF = 2/3
Do we have congruent included angles? Yes
So ΔABC ~ ΔDEF by SAS ~

AA
B
33°
A
E
36°
C
D
33°
36°
F
Here we have two sets of congruent angles
so, the triangles are similar by AA.
ΔABC ~ ΔDEF
Similarity of Triangles
Similar Triangles are symmetric, reflexive
and transitive:
Reflexive – ΔABC ~ ΔABC
Symmetric – If ΔABC ~ ΔDEF,
then ΔDEF ~ ΔABC.
Transitive – If ΔABC ~ ΔDEF and ΔDEF ~
ΔGHI, then ΔABC ~ ΔGHI
Common Example
D
E
A
B
Given: BE || CD
Prove: ΔABE ~ΔACD
C
ABE  ACD
A A
ΔABE ~ ΔACD by AA
Corresponding Angle
Theorem.
Reflexive Prop
Another Example
3
E
x
D
Solve for x and y.
1.5 y+2 ΔABE ~ ΔACD by AA
2 B 4 C
A
Because the two triangles are similar we can
set up the proportions.
Now
solve
for
x
&
y
AB
BE
AE


AC
CD
AD
2
1.5
3


6
y2
x3
x = 6 & y = 2.5
6.4 Parallel Lines and
Proportional Parts
Short Cuts
Side Splitter – If you have a triangle with a
segment parallel to one of the three sides, you
can use side splitter.
Converse of Side Splitter – If you have a
triangle where two sides are split proportionally,
then the segment is parallel to the sides.
Midsegment Theorem – If you go from the MP
of one side to the MP of the other side, then the
3rd side is 2x the segment.
Side Splitter
(Triangle Proportionality Theorem)
E
A
B
C
D Here we have a triangle
with a segment parallel
to a side – classic side
splitter case.
You don’t need to set up similar triangles and do
the proportions, just set up the proportion
AB/AE = BC/ED. Notice – you don’t use it for the
parallel sides, only the sides that are split!
The proof of this theorem is found on page 307
in the book if you care to see it.
Example
E
3
x
D Here we have a triangle
with a segment parallel
to a side – classic side
splitter case.
2 B 4 C
Just set up the proportion AB/AE = BC/ED.
A
2/3 = 4/x …….. solving for x we get x = 6
This is so much easier, but you have to be
careful that you’re only working with the sides
that have been split by parallel lines.
Converse of Side Splitter
E
6
D
3
2 B 4 C
Just set up the proportion AB/AE = BC/ED.
A
2/3 = 4/6 …….. Since this is true we can
conclude that BE and CD are parallel.
Midsegment Theorem
D
E
A
B
C
Since B is MP of AC and E
is MP of AD we can use
Midsegment Theorem
Since segment BE connects the two MP’s of
the sides, we can say that 2BE = CD or
BE = (½)CD
Remember this only works when we are
connecting the two MP’s of the two sides.
Corollaries
H
A
G
B
F
C
E
Here you have two
coplanar lines that
are cut by multiple
parallel lines.
D
AB/HG = BC/GF = CD/FE
AND
AB/HG =AC/HF = BD/GE = AD/HE
Continued
H
A
G
B
F
C
E
Here you have two
coplanar lines that
are cut by multiple
parallel lines.
D
Here if AB = BC = CD then HG = GF = FE.
6.5 Parts of Similar Triangles
Proportional Perimeters Theorem
C
12
A
F
20
12
B
18
30
Are these two Δ’s
similar?
Yes SSS~
D
18
E
What is the SF? SF is 2/3
FindProportional
the perimeters
of ΔABCTheorem
then ΔDEF.
Perimeters
–
two triangles
similar,
P ofIfΔABC
= 44, Pare
of ΔDEF
= then
66 the
to the
FindPerimeters
the ratio ofare
theproportional
corresponding
perimeters
measures
of
the
corresponding
sides.
It is the same as the SF -- 2/3
Special Segments of Δ’s
All the special segments of similar
triangles are proportional to the
corresponding sides (Same as the SF)
It works for all special segments, Altitudes,
Angle Bisectors, Perpendicular Bisectors
and Medians.
Angle Bisectors of Δ’s
Full Definition – An Angle Bisector of a
Triangle is a segment that is drawn from a
vertex to the opposite side that divides the
vertex angle into two congruent angles
and it divides the opposite side
proportionally.
Example is on the next slide.
Angle Bisectors of Δ’s
C
CD is an Angle Bisector
of ΔABC.
1 2
A
D
1 2
B
AC BC

AD BD
The side that is split by the
<bis is split proportionally.