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13.1 Ratio & Proportion
The student will learn about:
ratios, proportions,
similar triangles and
some special triangles.
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Ratios.
A ratio is the comparison of two numbers by
division. i.e. a/b.
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Proportions.
A proportion is a statement that two ratios are
equal. i.e.
a c

b d
a is the first term
b is the second term
c is the third term
d is the fourth term
a and d are the extremes.
b and c are the means.
d is the fourth proportion.
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Proportions.
If
a b

b c
Then b is called the geometric mean
between a and c and
b
ac
Not to be confused with the arithmetic mean.
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Geometric Mean.
a b
If 
then biscalledgeometricmeanbetweenaandc.
b c
It is easy to show that b = √(ac)
m CD = 3.99 cm
m CE = 5.99 cm
E
CF = 9.00 cm
b
a
D
4 6

6 9
c
C
 36  36
F
or 6 = √(4 · 9)
Construction of the geometric mean.
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Theorems.
a c
If 
then ad  bc. Multiply bothsidesby bd.
b d
a b
If ad  bc then  . Divide both sides by cd.
c d
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Theorems.
a c
ab cd
If 
then

. Add 1to bothsides.
b d
b
d
a c
ab cd
If  then

. Subtract 1from bothsides.
b d
b
d
These are merely the most useful of the
equations that may be derived from the
definition of proportion; there are many others.
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NOTE
We will need a proportionality
theorem and its converse for our work
on similar triangles.
Theorem
But first let’s look at the following relationship.
The two triangles have the same base and
altitudes, the lines are parallel, so they have the
same area.
Theorem
But first let’s look at the following relationship.
The two triangles have different bases and the
same altitudes, the lines are parallel. What is
the relationship of their areas?
The ratio of the areas is the same as
the ratio of the bases!
THEOREM: Triangles that have the same
altitudes have areas in proportion to their
bases.
C
h
A
D
B
1
h
AD
area ADC k ADC 2
AD



area DBC k DBC 1
DB
h DB
2
Now to the proportionality theorem
and its converse for our work on
similar triangles.
Basic Proportionality Theorem.
If a line parallel to one side of a triangle
intersects the other two sides, then it cuts off
segments which are proportional to these sides.
A
D
B
AB AC

AD AE
E
C
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If a line parallel to one side of a triangle intersects the
other two side, then it cuts off proportional segments.
What
given?
Given:isDE
∥ BC
(1) Construct BE and DC.
What will
Prove:
AB/AD
we prove?
= AC/AE
Construction
Why?
(2) Alt ∆BDE = alt ∆ADE
Bases & vertex.
Why?
Why?
Theorem
(4) Alt ∆ADE = alt ∆CDE
A
D
B
E
C
Bases and vertex.
Why?
Theorem
Why?
(6) k ∆BDE = k ∆CDE
Same
Why?bases & altitudes.
3,
5 & 6.
Why?
QED
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If a line parallel to one side of a triangle intersects the
other two side, then it cuts off proportional segments.
Given: DE ∥ BC
Prove: AB/AD = AC/AE
Previous slide.
Why?
Equals added
Why?
Substitution.
QED
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If a line parallel to one side of a triangle intersects the
other two side, then it cuts off proportional segments.
Given: DE ∥ BC
(7)
BD CE

AD AE
Prove: AB/AD = AC/AE
Previous slide.
Why?
Equals added
Why?
Substitution.
QED
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Converse of the Basic Proportionality
Theorem.
If a line intersects two sides of a triangle , and
cuts off segments proportional to these two
sides, the it is parallel to the third side.
A
AB AC

AD AE
D
B
E
C
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If a line intersects two sides of a triangle , and cuts off segments
proportional to these two sides, the it is parallel to the third side.
What
given? = AE/AC
Given:isAD/AB
(1) Let BC’ be parallel.
Prove:
DEwe∥ BC
What
will
prove?
By contradiction
Why?
(2) AD/AB = AE/AC’
(3) AD/AB = AE/AC
(4) AE/AC = AE/AC’
Why? theorem
Previous
Given
Why?
(5) C= C’
(6) → ←
Prop of proportions
Why?
Axiom
Why?
Unique parallel assumed
Why?
A
QED
D
B
E
C’
C
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Triangle Similarity
Definition. If the corresponding angles in two
triangles are congruent, and the sides are
proportional, then the triangles are similar.
A
D
E
F
B
C
AB AC BC


DE DF EF
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Basic Similarity Theorems
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AAA Similarity
Theorem. If the corresponding angles in two
triangles are congruent, then the triangles are
A
similar.
D
E
F
B
C
Since the angles are congruent we need to show
the corresponding sides are in proportion.
AB AC BC


DE DF EF
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If the corresponding angles in two triangles are
congruent, then the triangles are similar.
AB AC BC
Prove:
What
will
we
prove?

What
given?B=E, C=F
Given: is
A=D,
DE DF EF
Construction
Why?
(1) E’ so that AE’ = DE
(2) F’ so that AF’ = DF
(3) ∆AE’F’ ≌ ∆DEF
(4) AE’F =E =  B
Why?
Construction
SAS.
Why?
CPCTE & Given
Why?
(5) E’F’ ∥ BC
(6) AB/AE’ = AC /AF’
Corresponding angles
Why?
(7) AB/DE = AC /DF
Substitute
Why?
A
Why?
Prop Thm
(8) AC/DF = BC/EF is proven in the
same way.
QED
F’
E’
B
D
E
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F
AA Similarity
Theorem. If two corresponding angles in two
triangles are congruent, then the triangles are
similar.
A
D
E
F
B
C
In Euclidean geometry if you know two angles
you know the third angle.
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SAS Similarity
Theorem. If the two pairs of corresponding
sides are proportional, and the included angles
are congruent, then the triangles are similar.
A
D
E
F
B
C
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If the two pairs of corresponding sides are proportional, and the
included angles are congruent, then the triangles are similar.
we ~
prove?
Given: is
AB/DE
=AC/DF, A=D What
What
given?
Prove:will
∆ABC
∆DEF
Construction
Why?
(1) AE’ = DE, AF’ = DF
Why?
SAS
Given & substitution (1)
Why?
Basic Proportion Thm
Why?
(2) ∆AE’F’ ≌ ∆DEF
(3) AB/AE’ = AC/AF’
(4) E’F’∥ BC
(5) B =  AE’F’
(6) A =  A
(7) ∆ABC  ∆AE’F’
(8) ∆ABC  ∆DEF
A
QED
E’
B
Why?
Corresponding
angles
Reflexive
Why?
AA
Why?
Substitute 2 & 7
Why?
D
F’
E
C
F
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SSS Similarity
Theorem. If the corresponding sides are
proportional, then the triangles are similar.
A
D
E
F
Proof for homework.
B
C
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Right Triangle Similarity
Theorem. The altitude to the hypotenuse
separates the triangle into two triangles which are
similar to each other and to the original triangle.
b
a
c-x
A
Proof for homework.
B
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Two Special Triangles.
Ratios.
The ratio of the sides of a 30-60-90 triangle is
1 : √3 : 2
a
b
c


1
3 2
30
c
c
b
a
60
c
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Ratios.
The ratio of the sides of a 45-45-90 triangle is
1 : 1 : √2
a
a b
c
 
1 1
2
45
a
c
45
b=a
a
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QUIZ
In trapezoid ABCD we have AB = AD.
Prove that BD bisects ∠ ABC.
A
B
D
C
Summary.
• We learned about ratios.
• We learned about proportionality.
• We learned about the geometric means.
• We learned about the “Basic Proportionality
Theorem” and its converse.
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Summary.
• We learned about AAA similarity.
• We learned about SAS similarity.
• We learned about SSS similarity.
• We learned about similarity in right triangles.
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Assignment: 13.1